Etude du pont à poutres

February 15, 2017 | Author: Ghassen Zarrouk | Category: N/A
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ETUDE Caractéristiques transversales largeur roulable largeur de trottoir

Lr ltr

8 1.25

L d

21 0.5

Caractéristiques longitudinales Longueur de travée Distance d'about

Caractéristiques du béton des poutres Fc28

Béton Poids volumique (kN/m3) 25

(Mpa)

45

Module d'élasticité instantanné Ei 39125.83

Caractéristiques du béton de l’hourdis Béton Fc28

(Mpa)

25

Poids volumique (kN/m3) 25

Ei 32164.20

Caractéristiques de la poutre Poutre I

hp

hd

Poutre seule

1

0

Poutre +hourdis(Caractéristique mécaniques)

1.20

0.2

Poutre +hourdis(Caractéristique géométriques)

1.20

0.2

Conception de la section transversale Largeur total du pont Longeur de calcul Lc Hauteur de la poutre Avec Hourdis

Lt = Lr + 2 Ltr Lc=L-2 d Hp=Lc/17

10.5 20 1.20

Largeur de la table de compression

bt

0.44

largeur de l'âme au about Largeur de l’ame

btabout

première estimation

b0 (Lt – bt)/b0 =

6.707

b0corrigé

1.437

Section equivalente

b0équivalente

1.181

Inertie de la poutre

IGx

Nombre des éspacements

Nombre des poutres Hauteur de l’hourdis La largeur de la pré-dalle Diemsionnement de l'entretoise Le = he = epaisseur be =

0.3

ba

0.18 1.5

0.0789

Np

8

Hd=b0/16 b0 – bt + 0.1 =

0.2 1.097

10.06 m 1m 0.2 m

Détermination des paramètres fondamentaux Rigidité à la flexion

0.054867635

Le moment d'inertie de torsion

Senction en I Γ1 2(hp-hd)/ba = Γ2 Γ2 (bta-ba)/hta = Γ3 Γ3 (bta'-ba)/hta' = Γ3' Γ3'

0.0016 11.1111 0.0019 0.0004 1.4857 0.0003 0.0003 2.6 6.523E-05 6.576E-05

Formule Empirique Formule Sâada Formule Empirique Formule Sâada Formule Empirique Formule Sâada

γp

9.951E-04 x E (MN)

Rigidité à la torsion de la poutre et de l’entretoise γp =

9.951E-04 x E (MN)

γe = ρe =

0.000666667 x E (MN) 0.000666667 x E (MN)

Paramètres fondamentaux Paramètre de torsion

0.137379724 b= 0.791

Paramètre d'entretoisement

5.25

Calcul des coefficients de répartition transversale CRT Traçage des courbes K = F(e)

ρp= ρe=

Poutre de rive Y= Y'= e= e' = λ a' A b' B1 B2 E F σ ψ Χ β

0.05487 0.000667

b=

5.25

b0=

1.437

5.03 -5.03 -5.25 5.25

5.03 -5.03 -3.9375 3.9375

0.335 2.000 -9.527 0.147 -6.077 -6.077 8.452 -42.970 2.484 3.142 -3.010 -3.010

0.335 2.000 -14.798 0.147 8.827 -1.871 1.273 -29.092 2.484 2.356 -2.225 -3.010

C D Rβ Rψ Qβ Qψ

114.094 75.854 -26.610 -33.441 -66.864 70.856

62.600 29.487 -26.610 -5.038 -66.864 47.972

K0 K1 K

-0.261 0.130 -0.116

-0.358 0.186 -0.156

Poutre centrale 3 Y= Y'= e= e'=

3.593 -3.593 -5.250 5.250

3.593 -3.593 -3.938 3.938

λ a' A b' B1 B2 E F σ ψ Χ β C D Rβ Rψ Qβ Qψ

0.335 1.969 -9.527 1.105 -6.077 -6.077 0.381 -27.334 2.484 3.142 -2.150 -2.150 59.228 26.764 -1.201 -33.441 -42.533 70.856

0.335 1.969 -14.798 1.105 8.827 -1.871 0.057 -18.506 2.484 2.356 -1.365 -2.150 34.362 8.353 -1.201 -5.038 -42.533 47.972

K0 K1 K

-0.403 0.193 -0.182

-0.268 0.265 -0.071

Poutre centrale 2 Y= Y'= e= e'=

2.156 -2.156 -5.250 5.250

2.156 -2.156 -3.938 3.938

λ a' A

0.335 1.618 -9.527

0.335 1.618 -14.798

b' B1 B2 E F σ ψ Χ β C D Rβ Rψ Qβ Qψ

1.991 -6.077 -6.077 -2.173 -14.735 2.484 3.142 -1.290 -1.290 32.815 7.322 6.841 -33.441 -22.929 70.856

1.991 8.827 -1.871 -0.327 -9.976 2.484 2.356 -0.505 -1.290 22.633 0.973 6.841 -5.038 -22.929 47.972

K0 K1 K

-0.496 0.301 -0.201

-0.126 0.398 0.068

Poutre centrale 1 Y= Y'= e= e'=

0.719 -0.719 -5.250 5.250

0.719 -0.719 -3.938 3.938

λ a' A b' B1 B2 E F σ ψ Χ β C D Rβ Rψ Qβ Qψ

0.335 0.262 -9.527 2.502 -6.077 -6.077 -2.816 -4.614 2.484 3.142 -0.430 -0.430 22.166 0.701 8.866 -33.441 -7.179 70.856

0.335 0.262 -14.798 2.502 8.827 -1.871 -0.424 -3.124 2.484 2.356 0.355 -0.430 21.776 0.476 8.866 -5.038 -7.179 47.972

K0 K1 K

-0.412 0.492 -0.077

0.169 0.622 0.337

Détermination des CRT Caractéristiques du pont Nombre de dispositive de Retenue Dr Largeur chargeable Lch =Lr-0.5*Dr Nombre de voies Nv =ENT(Lch/3) Largeur d'une voie V =Lch/Nv Classe du pont Largeur d'une voie standard V0 a2 =V0/V

0 8m 2 4m

1 3.5 0.875

Poutre de rive Charge Al Nombre de voies chargées a1 =

1 1

x1=b-Ltr

-1 4 F (x1)=

x2=x1-V

-1 0 F (x2)=

Wal=F-1(x1)-F-1(x2) Kal=Wal/V CRT ηAL= Kal/Np a1 x ηAL x Lal =

5.658903303 1.414725826 0.176840728 0.707362913

Nombre de voies chargées a1=

2 1

x1=b-Ltr

4 F (x1)=

x2=x1-V

-1 -4 F (x2)=

-1

Wal=F-1(x1)-F-1(x2) Kal=Wal/(V*Nv) CRT ηAL= a1 x ηAL x Lal =

5.255953056 0.656994132 0.082124267 0.656994132

Recapitulation CRT 0.176840728

a1 x ηAL x Lal 0.707362913

Nombre de voies chargées 1

Charge qtr Nombre de trottoires chargées

1 -1 5.25 F (x1)=

x1=b

-1 4.000 F (x2)=

x2=b-Ltr -1

-1

Wtr=F (x1)-F (x2) Ktr=Wtr/(Ntr*Ltr) CRT ηtr=Ktr/Np Nombre de trottoires chargées

5.463956479 4.371165183 0.546395648 2

-1 5.25 F (x1)=

x1=b

-1

4.000 F (x2)=

x2=b-Ltr

-1 -5.25 F (x3)=

x3=-b

-1 -4.000 F (x4)=

x4=-b+Ltr -1

-1

-1

-1

Wtr=F (x1)+F (x4)-F (x2)-F (x3) Ktr=Wtr/(Ntr*Ltr) CRT ηtr=Ktr/Np

5.253479377 2.101391751 0.262673969

Recapitulation CRT 0.546395648

Ltr 1.25

Nbr de trottoires chargées 1

Charge Bc Nbr de file bc

1 1.2

x1 = x2 =

3.75 F(x1)= 1.75 F(x2)=

Kbc = ηbc = bc x ηbc =

2.059255483 0.257406935 0.308888323

Nbr de file bc

2 1.1

x1 = x2 = x3 = x4 =

3.75 1.75 1.25 -0.75

Kbc =0.5*[F(x1)+F(x2)+F(x3)+F(x4)] ηbc =Kbc/Np bc x ηbc =

F(x1)= F(x2)= F(x3)= F(x4)=

2.403676501 0.300459563 0.330505519

Recapitulation CRT 0.300459563

bc 1.1

Nbr de voies chargées 2

Charge Mc80 LMc80 = x1 = x2 = x3 = x4 = KMc80 =

0.85 m 4 3.15 1.2 0.35 1.682004698

F(x1)= F(x2)= F(x3)= F(x4)=

ηMc80 =

0.210250587

Recapitulation CRT 0.210

LMc80 0.85

Charge Mc120 LMc120 =

1m

x1 =b-Ltr x2 =x1-LMc120 x3 =x2-2.3 x4 =x3-1 KMc120 =(1/4)*[F(x1)+F(x2)+F(x3)+F(x4)] ηMc120 =KMc120/Np

4 3 0.7 -0.3 1.526727149 0.190840894

F(x1)= F(x2)= F(x3)= F(x4)=

Recapitulation CRT 0.191

LMc120 1

Récap des « CRT » : Poutre de rive Mc120 Mc80 Bc Trottoires AL Poutre centrale 3 Mc120 Mc80 Bc Trottoires AL

Poutre Modèle Mc120 Mc80

CRT 0.190840894 CRT 0.210250587 CRT 0.300459563 CRT 0.546395648 CRT 0.176840728

LMc120 1 LMc80 0.85 bc 1.1 Ltr 1.25 a1 x ηAL x Lal 0.707362913

CRT 0.170925756 CRT 0.178725872 CRT 0.357898401 CRT 0.073817684 CRT 0.150468608

LMc120 1 LMc80 0.85 bc 1.1 Ltr 1.25 a1 x ηAL x Lal 1.203748862

CRT 0.190840894 CRT 0.210250587

LMc120 1 LMc80 0.85

Bc Trottoires AL

CRT 0.357898401 CRT 0.546395648 CRT 0.176840728

bc 1.1 Ltr 1.25 a1 x ηAL x Lal 0.707362913

Calcul des sollicitations Charges permanentes Poids propre de la poutre Ba = Bc = Bmoyenne =

1 0.3385 m² 0.2515 m² 0.273 m²

Gpoutre = Poids propre de l’hourdis Gd = Poids de la super-structure Gétanchéité Gcouche de roulement Gtrottoirs Ggarde corps Poids de l'entretoise Ge = Gst =

6.817 KN/m

0.949 2.213 10.000 0.300 3.989 15.052

Gpermanent

29.054 KN/m

7.186 KN/m

γ

Moment fléchissant M

per x

(KN.m)

ELU ELS

x 1.35 1

γ

Effort tranchant Tperx (KN.m)

ELU ELS

Charge Al

1

Al = Poutre de rive A2 = qAL =

KN/m KN/m KN/m KN/m KN KN/m

x 1.35 1

13.55 KN/m² 11.85625 47.425 KN/m²

Moment fléchissant

γ

x

MALx (KN.m)

ELU ELS

γ

Effort tranchant T

Al x

(KN)

Poutre centrale 3 A2 = qAL =

1.6 1.2

ELU ELS

x 1.6 1.2

11.85625 94.85 KN/m²

γ

Moment fléchissant M

AL x

(KN.m)

ELU ELS

x 1.6 1.2

γ

Effort tranchant TALx (KN)

ELU ELS

Charge du trottoir

1

qtr = Qtr = Poutre de rive

x 1.6 1.2

1.5 kN/m² 1.875 kN/m

γ

Moment fléchissant M

Tr x

(KN.m)

ELU ELS

Tr x

(KN.m)

1.6 1

γ

Effort tranchant T

x

ELU ELS

x 1.6 1

Poutre centrale 3

γ

Moment fléchissant MTrx (KN.m)

ELU ELS

T

x

(KN.m)

1.6 1

γ

Effort tranchant Tr

x

ELU ELS

x 1.6 1

Charge Bc

1

Poids total G = SBc = SBt = SBr = Sb = δB =

2676.46 1320.00 640.00 100.00 1320.00 1.15

Poutre de rive Lc section choisit

KN KN KN KN KN

20 2

Lc/

Moment fléchissant M

section choisit Poids 1ER ESSIEU x1 x2 x3 x4 x5 x6 i xi yi Pi Yi Pi section choisit Poids 1ER ESSIEU x1 x2 x3 x4 x5 x6

Bc x

(KN.m)

ELU ELS

1.6 1.2

4 60 POIDS 2EM ESSIEU 5 6.5 11 15.5 17 21.5 1 5.00 3.75 120.00 450.00 6 60 POIDS 2EM ESSIEU 3.33 4.83 9.33 13.83 15.33 19.83

2 6.50 3.38 120.00 405.00

i xi yi Pi Yi Pi

1 3.33 2.78 120.00 333.33

section choisit Poids 1ER ESSIEU x1 x2 x3 x4 x5 x6

2 4.83 2.53 120.00 303.33

8 60 POIDS 2EM ESSIEU 2.5 4 8.5 13 14.5 19

i xi yi Pi Yi Pi

1 2.50 2.19 120.00 262.50

2 4.00 2.00 120.00 240.00

Poutre de rive

γ

Moment fléchissant MBcx (KN.m)

ELU ELS

x 1.6 1.2

Poutre Centrale 3

γ

Moment fléchissant MBcx (KN.m)

ELU ELS

x 1.6 1.2

Effort tranchant section choisit Poids 1ER ESSIEU x1 x2 x3 x4 x5 x6

0 60 POIDS 2EM ESSIEU 0 1.5 6 10.5 12 16.5

i xi yi

1 0.00 1.00

2 1.50 0.93

Pi Yi Pi section choisit Poids 1ER ESSIEU x1 x2 x3 x4 x5 x6 i xi yi Pi Yi Pi section choisit Poids 1ER ESSIEU x1 x2 x3 x4 x5 x6 i xi yi Pi Yi Pi section choisit Poids 1ER ESSIEU x1 x2 x3 x4 x5 x6 i xi yi Pi Yi Pi section choisit Poids 1ER ESSIEU

120.00 120.00

120.00 111.00

2 60 POIDS 2EM ESSIEU 10 11.5 16 20.5 22 26.5 1 10.00 0.50 120.00 60.00

2 11.50 0.43 120.00 51.00

4 60 POIDS 2EM ESSIEU 5 6.5 11 15.5 17 21.5 1 5.00 0.75 120.00 90.00

2 6.50 0.68 120.00 81.00

6 60 POIDS 2EM ESSIEU 3.33 4.83 9.33 13.83 15.33 19.83 1 3.33 0.83 120.00 100.00 8 60 POIDS 2EM ESSIEU

2 4.83 0.76 120.00 91.00

x1 x2 x3 x4 x5 x6

2.5 4 8.5 13 14.5 19

i xi yi Pi Yi Pi

1 2.50 0.88 120.00 105.00

2 4.00 0.80 120.00 96.00

Poutre de rive

γ

Effort tranchant TBcx (KN.m)

ELU ELS

x 1.6 1.2

Poutre Centrale 3

γ

Effort tranchant TBcx (KN.m)

ELU ELS

x 1.6 1.2

Charges militaires G= Poids du charge S = δMc =

2676.46 KN 720 KN 1.117809

Charge Mc80 Valeyr de la charge répartie q = Lc80

t y1 y2 y3 w

1 147 kN/m 4.9 m

=x(Lc-4.9)/Lc = x(Lc-x)/Lc = Y1*t/x = (Lc-t-4.9) Y1 /(Lc-x)

Poutre de rive

Moment fléchissant

γ

x

MMc80x (KN.m)

ELU ELS

1.35 1

Poutre Centrale 3

γ

Moment fléchissant MMc80x (KN.m)

γ

ELU ELS

x 1.35 1

x

y1 y2 w Poutre de rive

γ

Effort tranchant TMc80x (KN.m)

ELU ELS

x 1.35 1

Poutre Centrale 3

γ

Effort tranchant TMc80x (KN.m)

ELU ELS

Charge Mc120

1

Valeur de la charge répartie q = Lc120 = G= Poids du charge S = δMc =

γ

180 6.1 2676.46 1100 1.135904483

x 1.35 1

kN/m m KN KN

x

t y1 y2 y3 w

=x(Lc-4.9)/Lc = x(Lc-x)/Lc = Y1*t/x = (Lc-t-4.9) Y1 /(Lc-x)

Poutre de rive

γ

Moment fléchissant MMc120x (KN.m)

ELU

x 1.35

M

Mc120 x

(KN.m) ELS

1

Poutre Centrale 3

γ

Moment fléchissant MMc120x

ELU ELS

(KN.m)

γ

x 1.35 1

x

y1 y2 w Poutre de rive

γ

Effort tranchant TMc120x

(KN.m)

ELU ELS

x 1.35 1

Poutre Centrale 3

γ

Effort tranchant TMc120x

(KN.m)

ELU ELS

Sollicitations de calcul Poutre de rive Moment fléchissant Mx (KN.m)

ELU ELS

Poutre Centrale 3 Moment fléchissant Mx (KN.m)

ELU ELS

Poutre de rive Effort tranchant Tx (KN)

ELU

x 1.35 1

Tx (KN) ELS Poutre Centrale 3 Effort tranchant Tx (KN)

ELU ELS

Poutre Modèle Moment fléchissant Mx (KN.m)

ELU ELS

Effort tranchant Tx (KN)

Charge dominante

ELU ELS

Mc120

ETUDE DU PONT A POUTRES Réalisé par ZARROUK Ghassen Ingénieur Genie Civil de l'ENIT Veuillez demander un devis de prix par EMAIL: [email protected]

Module d'élasticité différé Ev

13041.94

Ev 10721.40

b0équivalente

ba

babout

bta(inférieur)

0

0.18

0.3

0.44

1.181

0.18

0.3

0.44

1.437

0.18

0.3

0.44

m m m

(Recommandation SETRA)

m m m m 7.000 m m m4

m m

x E (MN)

m

x E (MN) x E (MN) e=

5.35 L=

α=

0.137 θ =

20 0.791

5.03 -5.03 -2.625 2.625

5.03 -5.03 -1.3125 1.3125

5.03 -5.03 0 0

5.03 -5.03 1.3125 -1.3125

0.335 2.000 -19.045 0.147 26.061 0.349 -1.320 -17.473 2.484 1.571 -1.439 -3.010

0.335 2.000 -19.189 0.147 44.168 1.193 -2.121 -8.104 2.484 0.785 -0.654 -3.010

0.335 2.000 -9.418 0.147 57.791 1.251 -2.285 0.000 2.484 0.000 0.132 -3.010

0.335 2.000 18.687 0.147 55.119 0.956 -2.121 8.104 2.484 -0.785 0.917 -3.010

36.029 9.480 -26.610 5.225 -66.864 28.813

23.804 1.662 -26.610 8.394 -66.864 13.364

21.057 0.065 -26.610 9.039 -66.864 0.000

26.694 3.411 -26.610 8.394 -66.864 -13.364

-0.428 0.272 -0.169

-0.397 0.418 -0.095

-0.127 0.656 0.163

0.571 1.026 0.740

3.593 -3.593 -2.625 2.625

3.593 -3.593 -1.313 1.313

3.593 -3.593 0.000 0.000

3.593 -3.593 1.313 -1.313

0.335 1.969 -19.045 1.105 26.061 0.349 -0.060 -11.115 2.484 1.571 -0.579 -2.150 23.178 1.292 -1.201 5.225 -42.533 28.813

0.335 1.969 -19.189 1.105 44.168 1.193 -0.096 -5.155 2.484 0.785 0.206 -2.150 21.223 0.159 -1.201 8.394 -42.533 13.364

0.335 1.969 -9.418 1.105 57.791 1.251 -0.103 0.000 2.484 0.000 0.992 -2.150 27.717 4.045 -1.201 9.039 -42.533 0.000

0.335 1.969 18.687 1.105 55.119 0.956 -0.096 5.155 2.484 -0.785 1.777 -2.150 45.248 16.015 -1.201 8.394 -42.533 -13.364

-0.104 0.376 0.074

0.155 0.554 0.303

0.585 0.826 0.674

1.236 1.202 1.224

2.156 -2.156 -2.625 2.625

2.156 -2.156 -1.313 1.313

2.156 -2.156 0.000 0.000

2.156 -2.156 1.313 -1.313

0.335 1.618 -19.045

0.335 1.618 -19.189

0.335 1.618 -9.418

0.335 1.618 18.687

1.991 26.061 0.349 0.339 -5.992 2.484 1.571 0.281 -1.290 21.462 0.296 6.841 5.225 -22.929 28.813

1.991 44.168 1.193 0.545 -2.779 2.484 0.785 1.066 -1.290 28.837 4.749 6.841 8.394 -22.929 13.364

1.991 57.791 1.251 0.587 0.000 2.484 0.000 1.852 -1.290 47.693 17.827 6.841 9.039 -22.929 0.000

1.991 55.119 0.956 0.545 2.779 2.484 -0.785 2.637 -1.290 85.539 49.142 6.841 8.394 -22.929 -13.364

0.273 0.544 0.373

0.742 0.766 0.751

1.281 1.068 1.202

1.776 1.392 1.634

0.719 -0.719 -2.625 2.625

0.719 -0.719 -1.313 1.313

0.719 -0.719 0.000 0.000

0.719 0.719 1.313 1.313

0.335 0.262 -19.045 2.502 26.061 0.349 0.440 -1.876 2.484 1.571 1.141 -0.430 30.057 5.526 8.866 5.225 -7.179 28.813

0.335 0.262 -19.189 2.502 44.168 1.193 0.707 -0.870 2.484 0.785 1.926 -0.430 50.304 19.796 8.866 8.394 -7.179 13.364

0.335 0.262 -9.418 2.502 57.791 1.251 0.761 0.000 2.484 0.000 2.712 -0.430 90.581 53.695 8.866 9.039 -7.179 0.000

0.335 -3.099 -19.189 1.922 44.168 1.193 0.707 0.870 2.484 0.785 2.786 0.430 95.939 58.614 8.866 8.394 7.179 13.364

0.765 0.810 0.781

1.358 1.064 1.249

1.818 1.320 1.634

1.836 1.364 1.661

e

Poutre cen Charge Al Nombre de voies chargées a1 =

1 1

5.658903303

x1=

4

0

x2=

0

Wal= Kal= CRT ηAL= a1 x ηAL x Lal =

5.626799284 1.406699821 0.175837478 0.703349911

Nombre de voies chargées a1=

2 1

5.658903303

x1=

4

0.402950247

x2=

-4

Wal= Kal= CRT ηAL= a1 x ηAL x Lal =

9.629990893 1.203748862 0.150468608 1.203748862

Recapitulation Lal 4

a1 1

CRT 0.150468608

a1 x ηAL x Lal 1.203748862

Charge qtr Nombre de trottoires chargées

1

11.12285978

x1=

5.25

5.658903303

x2=

4

Wtr= Ktr= CRT ηtr= Nombre de trottoires chargées

0.738176841 0.590541473 0.073817684 2

11.12285978

x1=

5.25

5.658903303

x2=

4.000

0.613427348

x3=

-5.25

0.402950247

x4=

-4.000

Wtr= Ktr= CRT ηtr=

0.857460644 0.342984257 0.042873032

Recapitulation CRT 0.073817684

Ltr 1.25

Charge Bc

3.090550685 1.027960282

Nbr de file bc 1ér disposition x1 = x2 =

1 1.2 0.5 2.5

Kbc = ηbc = bc x ηbc =

3.090550685 1.027960282 0.702903145 -0.01406111

1.520 0.190 0.228

Nbr de file bc 1ér disposition x1 = x2 = x3 = x4 =

2 1.1 0.5 2.5 0 -2

Kbc = ηbc = bc x ηbc =

2.863 0.358 0.394

Recapitulation CRT 0.357898401

bc 1.1

Charge Mc80 3.434279643 2.339902555 0.674166693 0.279669903

LMc80 = 1ér disposition x1 = x2 = x3 = x4 = KMc80 =

0.85 0 0.85 2.8 3.65 1.429806976

ηMc80 =

0.178725872

Recapitulation CRT 0.179

LMc80 0.85

Charge Mc120 LMc120 = 1ér disposition x1 = x2 = x3 = x4 = KMc120 = ηMc120 =

3.434279643 2.169650829 0.421630041 0.081348084

1 2.5 1.5 -0.8 -1.8 1.367 0.171

Recapitulation CRT 0.171

Nbr de voies chargées 2 Nbr de trottoires chargées 1 Nombre de voies chargées 1

Lal 4

a1 1

Nbr de voies chargées 2 Nbr de trottoires chargées 1 Nombre de voies chargées 2

Lal 8

a1 1

LMc120 1

Nbr de voies chargées 2 Nbr de trottoires chargées 1 Nombre de voies chargées 1

LAl 4

Lt Lc

a1 1

21 20

Coefficients de majoration 1.2 1.4 1.05 1.05

x=Lc /2 10 1961.15 1452.70

x = Lc/4 5 1470.86 1089.53

x=Lc/6 3.33 1089.53 807.06

x=Lc/8 2.5 858.00 635.56

x=Lc /2 10.00 0.00 0.00

x = Lc/4 5.00 196.12 145.27

x=Lc/6 3.33 261.49 193.69

x=Lc/8 2.50 294.17 217.91

x=Lc /2 10

x = Lc/4 5

x=Lc/6 3.33

x=Lc/8 2.5

670.93 503.20

503.20 377.40

372.74 279.56

293.53 220.15

x=Lc /2 10.00 46.21 34.66

x = Lc/4 5.00 87.09 65.31

x=Lc/6 3.33 102.18 76.64

x=Lc/8 2.50 109.97 82.47

x=Lc /2 10 1141.76 856.32

x = Lc/4 5 856.32 642.24

x=Lc/6 3.33 634.31 475.73

x=Lc/8 2.5 499.52 374.64

x=Lc /2 10.00 78.63 58.97

x = Lc/4 5.00 148.20 111.15

x=Lc/6 3.33 173.89 130.41

x=Lc/8 2.50 187.13 140.35

x=Lc /2 10 81.96 51.22

x = Lc/4 5 61.47 38.42

x=Lc/6 3.33 45.53 28.46

x=Lc/8 2.5 35.86 22.41

x=Lc /2 10.00 4.10 2.56

x = Lc/4 5.00 9.22 5.76

x=Lc/6 3.33 11.38 7.11

x=Lc/8 2.50 12.55 7.84

x=Lc /2 10 11.07 6.92

x = Lc/4 5 8.30 5.19

x=Lc/6 3.33 6.15 3.84

x=Lc/8 2.5 4.84 3.03

x=Lc /2 10.00 0.55 0.35

x = Lc/4 5.00 1.25 0.78

x=Lc/6 3.33 1.54 0.96

x=Lc/8 2.50 1.70 1.06

Poutre centrale section choisit

Lc/

2

Moment fléchissant 917.56 688.17

M

Bc x

ELU ELS

(KN.m)

120 y1 y2 y3 y4 y5 y6

3.75 3.375 2.250 1.125 0.750 -0.375 3 11.00 2.25 60.00 135.00

4 15.50 1.13 120.00 135.00

5 17.00 0.75 120.00 90.00

6 21.50 0.00 60.00 0.00 MAX=

120 y1 y2 y3 y4 y5 y6

2.78 2.528 1.778 1.028 0.778 0.028

3 9.33 1.78 60.00 106.67

4 13.83 1.03 120.00 123.33

5 15.33 0.78 120.00 93.33

6 19.83 0.03 60.00 1.67 MAX=

120 y1 y2 y3 y4 y5 y6

2.19 2.000 1.438 0.875 0.688 0.125 3 8.50 1.44 60.00 86.25

4 13.00 0.88 120.00 105.00

5 14.50 0.69 120.00 82.50

6 19.00 0.13 60.00 7.50 MAX=

x=Lc /2 10 917.56 688.17

x = Lc/4 5 749.85 562.39

x=Lc/6 3.33 582.71 437.03

x=Lc/8 2.5 474.91 356.18

x=Lc /2 10 1092.96 819.72

x = Lc/4 5 893.20 669.90

x=Lc/6 3.33 694.11 520.58

x=Lc/8 2.5 565.69 424.27

120 y1 y2 y3 y4 y5 y6

1.00 0.925 0.700 0.475 0.400 0.175 3 6.00 0.70

4 10.50 0.48

5 12.00 0.40

6 16.50 0.18

60.00 42.00

120.00 57.00

120.00 48.00

60.00 10.50

5 22.00 0.00 120.00 0.00

6 26.50 0.00 60.00 0.00

5 17.00 0.15 120.00 18.00

6 21.50 0.00 60.00 0.00

5 15.33 0.23 120.00 28.00

6 19.83 0.01 60.00 0.50

120 y1 y2 y3 y4 y5 y6

0.50 0.425 0.200 -0.025 -0.100 -0.325 3 16.00 0.20 60.00 12.00

4 20.50 0.00 120.00 0.00

120 y1 y2 y3 y4 y5 y6

0.75 0.675 0.450 0.225 0.150 -0.075 3 11.00 0.45 60.00 27.00

4 15.50 0.23 120.00 27.00

120 y1 y2 y3 y4 y5 y6

0.83 0.758 0.533 0.308 0.233 0.008 3 9.33 0.53 60.00 32.00

120

4 13.83 0.31 120.00 37.00

y1 y2 y3 y4 y5 y6

0.88 0.800 0.575 0.350 0.275 0.050 3 8.50 0.58 60.00 34.50

4 13.00 0.35 120.00 42.00

5 14.50 0.28 120.00 33.00

6 19.00 0.05 60.00 3.00

x=Lc /2 10 74.53 55.90

x = Lc/4 5 147.24 110.43

x=Lc/6 3.33 174.81 131.11

x=Lc/8 2.5 189.96 142.47

x=Lc /2 10 88.78 66.58

x = Lc/4 5 175.39 131.54

x=Lc/6 3.33 208.23 156.17

x=Lc/8 2.5 226.28 169.71

x=Lc /2 10

x = Lc/4 5

x=Lc/6 3.33

x=Lc/8 2.5

7.55 5.00 3.78 3.78 21.50

x=Lc /2 10

3.78 3.75 2.83 2.83 16.12

x = Lc/4 5

2.52 2.78 2.10 2.10 11.94

x=Lc/6 3.33

1.89 2.19 1.65 1.65 9.41

x=Lc/8 2.5

1002.70 742.74

752.02 557.05

557.05 412.63

438.68 324.95

x=Lc /2 10 852.35 631.37

x = Lc/4 5 639.26 473.53

x=Lc/6 3.33 473.53 350.76

x=Lc/8 2.5 372.90 276.23

x=Lc /2 10

x = Lc/4 5

x=Lc/6 3.33

x=Lc/8 2.5

0.50 0.26 1.85

0.75 0.51 3.07

0.83 0.59 3.48

0.88 0.63 3.69

x=Lc /2 10 86.27 63.91

x = Lc/4 5 143.41 106.23

x=Lc/6 3.33 162.45 120.33

x=Lc/8 2.5 171.97 127.39

x=Lc /2 10 73.34 54.32

x = Lc/4 5 121.90 90.30

x=Lc/6 3.33 138.09 102.29

x=Lc/8 2.5 146.19 108.29

x=Lc /2 10

x = Lc/4 5 3.475 3.75 2.61 2.61 19.39

x=Lc/6 3.33

x=Lc/8 2.5

x = Lc/4 5 1021.22

x=Lc/6 3.33 756.46

6.95 5 3.48 3.48 25.85

x=Lc /2 10 1361.63

2.32 2.78 1.93 1.93 14.36

1.74 2.19 1.52 1.52 11.31

x=Lc/8 2.5 595.71

1008.61

756.46

560.34

441.27

x=Lc /2 10 1219.54 903.36

x = Lc/4 5 914.65 677.52

x=Lc/6 3.33 677.52 501.87

x=Lc/8 2.5 533.55 395.22

x=Lc /2 10

x = Lc/4 5

x=Lc/6 3.33

x=Lc/8 2.5

0.50 0.20 2.12

0.75 0.45 3.64

0.83 0.53 4.15

0.88 0.57 4.41

x=Lc /2 10 111.66 82.71

x = Lc/4 5 191.99 142.22

x=Lc/6 3.333 218.77 162.05

x=Lc/8 2.5 232.16 171.97

x=Lc /2 10 100.01 74.08

x = Lc/4 5 171.96 127.38

x=Lc/6 3.33 195.94 145.14

x=Lc/8 2.5 207.93 154.02

x=Lc /2 10 3322.78 2461.32

x = Lc/4 5 2492.09 1845.99

x=Lc/6 3.33 1845.99 1367.40

x=Lc/8 2.5 1453.72 1076.83

x=Lc /2 10 3180.69 2356.06

x = Lc/4 5 2385.52 1767.05

x=Lc/6 3.33 1789.79 1331.48

x=Lc/8 2.5 1428.54 1062.86

x=Lc /2 10 111.66

x = Lc/4 5 388.11

x=Lc/6 3.33 480.26

x=Lc/8 2.5 526.33

82.71

287.49

355.75

389.88

x=Lc /2 10 100.01 74.08

x = Lc/4 5 372.75 277.59

x=Lc/6 3.33 471.26 350.83

x=Lc/8 2.5 522.15 388.67

x=Lc /2 10 3322.78 2461.32

x = Lc/4 5 2492.09 1845.99

x=Lc/6 3.33 1845.99 1367.40

x=Lc/8 2.5 1459.55 1082.24

x=Lc /2 10 111.66 82.71

x = Lc/4 5 388.11 287.49

x=Lc/6 3.33 481.10 356.98

x=Lc/8 2.5 533.00 395.46

OUTRES

nie Civil de l'ENIT

[email protected]

bta'(superieur)

h1

h2

0.44

0.05

0.15

0.44

0.05

0.15

0.44

0.05

0.15

bta'

h'2 h'1

ba babout

h1 h2 bta

5.03 -5.03 2.625 -2.625

5.03 -5.03 3.9375 -3.9375

5.03 5.03 5.25 5.25

0.335 2.000 74.493 0.147 16.146 0.564 -1.320 17.473 2.484 -1.571 1.702 -3.010

0.335 2.000 163.899 0.147 -86.430 0.191 1.273 29.092 2.484 -2.356 2.488 -3.010

0.335 -29.822 -9.527 -19.453 -6.077 -6.077 8.452 42.970 2.484 3.142 3.010 3.010

42.960 14.349 -26.610 5.225 -66.864 -28.813

76.330 41.032 -26.610 -5.038 -66.864 -47.972

114.094 75.854 -26.610 -33.441 66.864 70.856

1.896 1.569 1.775

3.945 2.302 3.336

6.516 3.143 5.266

3.593 -3.593 2.625 -2.625

3.593 3.593 3.938 3.938

3.593 3.593 5.250 5.250

0.335 1.969 74.493 1.105 16.146 0.564 -0.060 11.115 2.484 -1.571 2.562 -2.150 80.794 44.929 -1.201 5.225 -42.533 -28.813

0.335 -18.995 -14.798 -7.686 8.827 -1.871 0.057 18.506 2.484 2.356 2.935 2.150 107.675 69.663 -1.201 -5.038 42.533 47.972

0.335 -18.995 -9.527 -7.686 -6.077 -6.077 0.381 27.334 2.484 3.142 2.150 2.150 59.228 26.764 -1.201 -33.441 42.533 70.856

2.067 1.645 1.910

2.849 1.983 2.528

3.435 2.110 2.944

2.156 2.156 2.625 2.625

2.156 2.156 3.938 3.938

2.156 2.156 5.250 5.250

0.335 -9.447 -19.045

0.335 -9.447 -14.798

0.335 -9.447 -9.527

-0.961 26.061 0.349 0.339 5.992 2.484 1.571 2.861 1.290 101.630 63.927 6.841 5.225 22.929 28.813

-0.961 8.827 -1.871 -0.327 9.976 2.484 2.356 2.075 1.290 56.062 24.251 6.841 -5.038 22.929 47.972

-0.961 -6.077 -6.077 -2.173 14.735 2.484 3.142 1.290 1.290 32.815 7.322 6.841 -33.441 22.929 70.856

1.935 1.544 1.790

1.666 1.453 1.587

1.273 1.334 1.296

0.719 0.719 2.625 2.625

0.719 0.719 3.938 3.938

0.719 0.719 5.250 5.250

0.335 -3.099 -19.045 1.922 26.061 0.349 0.440 1.876 2.484 1.571 2.001 0.430 53.091 21.932 8.866 5.225 7.179 28.813

0.335 -3.099 -14.798 1.922 8.827 -1.871 -0.424 3.124 2.484 2.356 1.215 0.430 31.381 6.382 8.866 -5.038 7.179 47.972

0.335 -3.099 -9.527 1.922 -6.077 -6.077 -2.816 4.614 2.484 3.142 0.430 0.430 22.166 0.701 8.866 -33.441 7.179 70.856

1.374 1.173 1.300

0.742 0.971 0.827

0.077 0.815 0.351

Poutre centrale 3

F(x1)=

5.626799284

F(x2)=

0

F(x1)=

5.626799284

F(x2)=

-4.003191609

Nombre de voies chargées 2

Lal 8

a1 1

F(x1)=

6.364976126

F(x2)=

5.626799284

F(x1)=

6.364976126

F(x2)=

5.626799284

F(x3)=

-4.122475411

F(x4)=

-4.003191609

Nbr de trottoires chargées 1

F(x1)= F(x2)= Total

2éme disposition 1.696 x'1= 1.344 x'2= 3.041

F(x1)= F(x2)= F(x3)= F(x4)= Total

1.696 1.344 1.634 0.999 5.674

2éme disposition x'1 = x'2 = x'3 = x'4 =

Nbr de voies chargées 2

m F(x1)= F(x2)= F(x3)= F(x4)=

1.634 1.702 1.236 0.926 5.498

2éme disposition x1 = x2 = x3 = x4 =

m F(x1)= F(x2)= F(x3)= F(x4)=

1.344 1.629 1.425 1.072 5.470

2éme disposition x1 = x2 = x3 = x4 =

0 0 0.00 0.00 0.00 0.00 397.61 294.53

0 0

0.00 0.00 0.00 0.00 134.19 100.64

0 0 0.00 0.00 0.00 0.00 228.35 171.26

0 0 0.00 0.00 0.00 0.00 16.39 10.24

0 0 0.00 0.00 0.00 0.00 2.21 1.38

20 2

ment fléchissant 1092.96 819.72

1.6 1.2

somme

section choisit Poids 1ER ESSIEU x1 x2 x3 x4 x5 x6

4 60 3.5 5 9.5 14 15.5 20

i xi yi Pi 1215.00 Yi Pi 1237.50 section choisit Poids 1ER ESSIEU x1 x2 x3 x4 x5 x6

1 3.5 2.625 120 315 6 60 1.83 3.33 7.83 12.33 13.83 18.33

somme

somme

0 0 0.00 0.00

0 0 0.00 0.00

i xi yi Pi 961.67 Yi Pi 961.67 section choisit Poids 1ER ESSIEU x1 x2 x3 x4 x5 x6 i xi yi Pi 783.75 Yi Pi 783.75

1 1.83 1.53 120.00 183.33 8 60 1 2.5 7 11.5 13 17.5 1 1 0.875 120 105

somme 388.50

somme 123.00

somme 243.00

somme 288.50

somme 313.50

0 0 235.41 176.56

0 0 280.41 210.31

0 0 0.00 0.00 0.00 0.00 0.00

0 0

0.00 0.00

0 0 0.00 0.00

0 0 1.00 0.76 4.30

0 0 200.54 148.55

0 0 170.47 126.27

0 0 0 0 0.00 0.00 0.00

0 0 0.00

0.00

0 0 0.00 0.00

0 0 1.00 0.70 5.17

0 0 272.33 201.72

0 0 243.91 180.67

0 0 0.00 0.00

0 0 0.00 0.00

0 0 669.94

496.25

0 0 680.24 506.22

0 0 0.00 0.00 0 0 694.42 515.08

hta(inférieur)

h'1

h'2

hta'(superieur)

0.175

0.04

0.08

0.1

0.175

0.04

0.08

0.1

0.175

0.04

0.08

0.1

POUTRE DE RIVE 6.000 5.000 4.000 3.000

2.000 1.000 0.000 -6

-4

-2

-1.000

0

FONCTION Polynomiale de LEGENDRE x6 y= x= y(x)=

x5

-1.26135E-05

-9.92641E-05

0 0.163143829

PRIMITIVE x7 Y-1= x= y-1(x)=

x6 -1.80193E-06 0 0

-1.6544E-05

POUTRE CENTRALE 3 3.500 3.000 2.500 2.000 1.500 1.000 0.500

-6.000

-4.000

-2.000

0.000 0.000 -0.500

2.000

POUTRE CENTRALE 2 2.000 1.500 1.000

4.000

1.000 0.500

-6.000

-4.000

0.000 0.000

-2.000

2.000

-0.500

POUTRE CENTRAL 1 1.800 1.600 1.400 1.200 1.000

0.800 0.600 0.400 0.200 -6.000

-4.000

0.000 -0.2000.000

-2.000

FONCTION Polynomiale de LEGENDRE x8 y= x= y(x)=

2.000

x7

1.81756E-06

-1.5037E-05

0 1.633555407

PRIMITIVE x9 Y-1= x= y-1(x)=

x8 2.01951E-07 0

-1.87962E-06

1 F(x'1)= 3 F(x'2)= Total

1.694996184 1.162324116 2.857

0.8 2.8 0.3 -1.7

F(x'1)= F(x'2)= F(x'3)= F(x'4)= Total

1.703528993 1.236211345 1.678655293 1.107978783 5.726

0.5 1.35 3.3 4.15

F(x1)= F(x2)= F(x3)= F(x4)=

1.696428522 1.65509036 1.051569302 0.75531497 5.158

3éme disposition x1 = x2 = x3 = x4 =

2 1 -1.3 -2.3

F(x1)= F(x2)= F(x3)= F(x4)=

1.50646986 1.694996184 1.253355242 0.892924099 5.348

3éme disposition x1 = x2 = x3 = x4 =

POIDS 2EM ESSIEU

120 y1 y2 y3 y4 y5 y6

2.625 3.75 2.625 1.5 1.125 0

2 5 3.75 120 450

3 9.5 2.625 60 157.5

POIDS 2EM ESSIEU

120 y1 y2 y3 y4 y5 y6

4 14 1.50 120 180

1.53 2.78 2.03 1.28 1.03 0.28

5 15.5 1.13 120 135

2 3.33 2.78 120 333.33

3 7.83 2.03 60 121.67

POIDS 2EM ESSIEU

120 y1 y2 y3 y4 y5 y6

2 2.5 2.1875 120 262.5

4 12.33 1.28 120 153.33

5 13.83 1.03 120 123.33

0.875 2.1875 1.625 1.0625 0.875 0.3125 3 7 1.625 60 97.5

4 11.5 1.06 120 127.5

5 13 0.88 120 105

y2

y1

0.5281

0.4719

0.4238

0.7762

0.3931

0.8069

Ip(médiane) 0.0180 0.0789 0.0836

OUTRE DE RIVE

2

4

6

x4

x3

x2

0.00018884

0.01100223

0.092167949

x5

x4

x3

3.7768E-05

0.002750557

POUTRE CENTRALE 3

4.000

6.000

0.03072265

POUTRE CENTRALE 2

4.000

6.000

RE CENTRAL 1 POUTRE CENTRAL 1

4.000

6.000

x6

x5

x4

-0.00013881

0.000871069

0.00450588

x7 x6 -1.9829E-05

x5 0.000145178

0.000901176

-0.5 0.35 2.3 3.15

F(x1)= F(x2)= F(x3)= F(x4)=

1.515692316 1.684051975 1.412708473 1.106775141 5.719

1.5 0.5 -1.8 -2.8

F(x1)= F(x2)= F(x3)= F(x4)=

1.628769198 1.696428522 1.071543736 0.72275484 5.119

6 20 0.00 60 somme 0

1237.5

6 18.33 0.28 60 somme 16.67

931.67

6 17.5 0.31 60 somme 18.75

716.25

Ip(about) 0.0222 0.0876 0.0937

Surface au mediane (m²) 0.2515 0.4878 0.5389

x

1

0.299318542

0.163143829

x2

x 0.149659271

0.163143829

x3

x2

-0.017824495

-0.111097775

x4

x3 -0.004456124

-0.037032592

Surface au about (m²) 0.3385 0.5748 0.6259

x

1

0.185138123

1.63355541

x2

x 0.092569062

1.63355541

Interpolation polynomiale de Legendre Points de passage

Le polynome a pour équation

x

y

-5.250

-0.116

0.16314

-3.938

-0.156

0.29932

.x

-2.625

-0.169

0.09217

.x2

-1.313

-0.095

0.01100

.x3

0.000

0.163

0.00019

.x4

1.313

0.740

-0.00010

.x5

2.625

1.775

-0.00001

.x6

3.938

3.336

0.000

.x7

5.250

5.266

0.000

.x8

Le polynome de degré 8 passe par le point x0 y

5.250

#REF! -5 -4 -3 -2 -1 0 1 2 3 4 5

x0

y 1.312 0.73945929

-0.173389522 -0.165261304 -0.172138497 -0.149448673 -0.054733503 0.163143829 0.565709512 1.217508271 2.169650829 3.434279643 4.949952944

Interpolation polynomiale de Legendre

6 5 4 3

Concordence Original

2 1 0 -6

-4

-2

0

2

4

6

-1

e= k=

Concordence Original

#REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF!

1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

1.00 0.56 0.25 0.06 0.00 0.06 0.25 0.56 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

-1.00 -0.42 -0.13 -0.02 0.00 0.02 0.13 0.42 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

-2.625 -1.3125 0 1.3125 -0.168699747 -0.09492686 0.16314383 0.73975349

2.625 1.7747948

8

5.250

-5.25 -3.9375 -0.11584786 -0.15632145

#REF!

1.00 0.32 0.06 0.00 0.00 0.00 0.06 0.32 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

-1.00 -0.24 -0.03 0.00 0.00 0.00 0.03 0.24 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

3.9375 5.25 3.33573974 5.26594393

1.00 0.18 0.02 0.00 0.00 0.00 0.02 0.18 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

-1.00 -0.13 -0.01 0.00 0.00 0.00 0.01 0.13 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

1.00 0.10 0.00 0.00 0.00 0.00 0.00 0.10 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00

-0.12 -0.16 -0.17 -0.09 0.16 0.74 1.77 3.34 5.27 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.163 1.571 2.540 1.592 0.143 -0.396 -0.264 -0.077 -0.008 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.16314383 0.29931854 0.09216795 0.01100223 nm 0.00018884 -9.9264E-05 -1.2613E-05 -6.9754E-07 -1.3552E-08 0 0 0 0 0 0 0 0 0 0 0

5.250 -5.250 10.500 5.250

-1.26 -1.155 -1.05 -0.945 -0.84 -0.735 -0.63 -0.525 -0.42 -0.315 -0.21 -0.105 8.4655E-15 0.105 0.21 0.315 0.42 0.525 0.63 0.735 0.84 0.945 1.05 1.155 1.26 1.365 1.47 1.575 1.68 1.785 1.89 1.995 2.1 2.205 2.31 2.415 2.52 2.625 2.73 2.835 2.94 3.045 3.15 3.255 3.36

3.465 3.57 3.675 3.78 3.885 3.99 4.095 4.2 4.305 4.41 4.515 4.62 4.725 4.83 4.935 5.04 5.145 5.25 5.355 5.46 5.565 5.67 5.775 5.88 5.985 6.09 6.195 6.3

-6.3 -6.195 -6.09 -5.985 -5.88 -5.775 -5.67 -5.565 -5.46 -5.355 -5.25 -5.145 -5.04 -4.935 -4.83 -4.725 -4.62 -4.515 -4.41 -4.305 -4.2 -4.095 -3.99 -3.885 -3.78 -3.675 -3.57 -3.465 -3.36 -3.255 -3.15 -3.045 -2.94 -2.835 -2.73 -2.625 -2.52 -2.415 -2.31 -2.205 -2.1 -1.995 -1.89 -1.785 -1.68 -1.575

-0.32151957 -0.29883395 -0.27860774 -0.26065072 -0.24478378 -0.23083837 -0.21865596 -0.20808751 -0.19899297 -0.19124079 -0.1847074 -0.17927677 -0.17483992 -0.17129449 -0.1685443 -0.16649891 -0.16507322 -0.16418704 -0.16376476 -0.1637349 -0.16402981 -0.16458528 -0.16534018 -0.16623619 -0.16721743 -0.16823021 -0.16922265 -0.1701445 -0.17094682 -0.1715817 -0.1720021 -0.17216151 -0.17201382 -0.17151308 -0.1706133 -0.16926826 -0.16743138 -0.1650555 -0.16209279 -0.15849459 -0.15421129 -0.14919223 -0.14338557 -0.13673825 -0.1291959 -0.12070276

-1.47 -1.365 -1.26 -1.155 -1.05 -0.945 -0.84 -0.735 -0.63 -0.525 -0.42 -0.315 -0.21 -0.105 8.4655E-15 0.105 0.21 0.315 0.42 0.525 0.63 0.735 0.84 0.945 1.05 1.155 1.26 1.365 1.47 1.575 1.68 1.785 1.89 1.995 2.1 2.205 2.31 2.415 2.52 2.625 2.73 2.835 2.94 3.045 3.15 3.255 3.36

-0.11120167 -0.10063397 -0.08893956 -0.07605681 -0.06192261 -0.04647236 -0.02964001 -0.01135806 0.00844234 0.02983137 0.05288044 0.07766212 0.10425006 0.13271882 0.16314383 0.19560119 0.23016755 0.26691996 0.30593569 0.34729203 0.39106615 0.43733482 0.48617425 0.53765982 0.59186586 0.6488654 0.70872987 0.77152886 0.83732979 0.90619763 0.97819459 1.05337976 1.1318088 1.21353358 1.29860179 1.38705662 1.47893629 1.57427372 1.6730961 1.77542442 1.88127308 1.99064944 2.10355333 2.21997658 2.33990255 2.46330563 2.59015071

3.465 3.57 3.675 3.78 3.885 3.99 4.095 4.2 4.305 4.41 4.515 4.62 4.725 4.83 4.935 5.04 5.145 5.25 5.355 5.46 5.565 5.67 5.775 5.88 5.985 6.09 6.195 6.3

2.72039266 2.85397582 2.99083344 3.13088709 3.27404614 3.42020713 3.5692532 3.72105348 3.87546247 4.03231937 4.19144751 4.35265361 4.51572718 4.68043979 4.8465444 5.01377466 5.18184417 5.35044576 5.51925076 5.68790821 5.85604416 6.02326081 6.18913578 6.35322127 6.51504327 6.67410073 6.82986468 6.98177746

Interpolation polynomiale de Legendre Points de passage

Le polynome a pour équation

x

y

-5.250

-0.077

1.63356

-3.938

0.337

0.18514

.x

-2.625

0.781

-0.11110

.x2

-1.313

1.249

-0.01782

.x3

0.000

1.634

0.00451

.x4

1.313

1.661

0.00087

.x5

2.625

1.300

-0.00014

.x6

3.938

0.827

-0.000015

.x7

5.250

0.351

0.000002

.x8

Le polynome de degré 8 passe par le point x0 y

x0

5.250

#REF!

y 1.312 1.66099358

-5.25 -5 -4 -3 -2 -1 0 1 2 3 4 5 5.25

-0.07683749 -0.03152332 0.31467038 0.65645052 0.99921036 1.35865686 1.63355541 1.69499618 1.50646986 1.16232412 0.80545659 0.45838373 0.35087393

Interpolation polynomiale de Legendre

our équation

1.8 1.6 1.4 1.2 1 Original

0.8

Concordence

0.6 0.4 0.2 0 -6

-4

-2

-0.2

0

2

4

6

e= k=

Original Concordence

#REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF!

1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

1.00 0.56 0.25 0.06 0.00 0.06 0.25 0.56 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

-1.00 -0.42 -0.13 -0.02 0.00 0.02 0.13 0.42 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

8

5.250

-5.25 -3.9375 -0.07683749 0.33728028

#REF!

-2.625 -1.3125 0 1.3125 2.625 0.781415729 1.24886527 1.63355541 1.66083463 1.29988208

1.00 0.32 0.06 0.00 0.00 0.00 0.06 0.32 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

-1.00 -0.24 -0.03 0.00 0.00 0.00 0.03 0.24 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

3.9375 5.25 0.82655681 0.35087393

1.00 0.18 0.02 0.00 0.00 0.00 0.02 0.18 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

-1.00 -0.13 -0.01 0.00 0.00 0.00 0.01 0.13 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

1.00 0.10 0.00 0.00 0.00 0.00 0.00 0.10 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00

-0.08 0.34 0.78 1.25 1.63 1.66 1.30 0.83 0.35 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

1.634 0.972 -3.062 -2.579 3.423 3.474 -2.906 -1.653 1.049 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

1.63355541 0.18513812 -0.11109777 -0.0178245 nm 0.00450588 0.00087107 -0.00013881 -1.5037E-05 1.8176E-06 0 0 0 0 0 0 0 0 0 0 0

5.250 -5.250 10.500 5.250

-1.26 -1.155 -1.05 -0.945 -0.84 -0.735 -0.63 -0.525 -0.42 -0.315 -0.21 -0.105 8.4655E-15 0.105 0.21 0.315 0.42 0.525 0.63 0.735 0.84 0.945 1.05 1.155 1.26 1.365 1.47 1.575 1.68 1.785 1.89 1.995 2.1 2.205 2.31 2.415 2.52 2.625 2.73 2.835 2.94 3.045 3.15 3.255 3.36

3.465 3.57 3.675 3.78 3.885 3.99 4.095 4.2 4.305 4.41 4.515 4.62 4.725 4.83 4.935 5.04 5.145 5.25 5.355 5.46 5.565 5.67 5.775 5.88 5.985 6.09 6.195 6.3

-6.3 -6.195 -6.09 -5.985 -5.88 -5.775 -5.67 -5.565 -5.46 -5.355 -5.25 -5.145 -5.04 -4.935 -4.83 -4.725 -4.62 -4.515 -4.41 -4.305 -4.2 -4.095 -3.99 -3.885 -3.78 -3.675 -3.57 -3.465 -3.36 -3.255 -3.15 -3.045 -2.94 -2.835 -2.73 -2.625 -2.52 -2.415 -2.31 -2.205 -2.1 -1.995 -1.89 -1.785 -1.68 -1.575

0.72276479 0.51171623 0.34213061 0.2084753 0.10578587 0.02962317 -0.02396737 -0.05849424 -0.07705627 -0.08237708 -0.07683749 -0.06250614 -0.0411682 -0.01435241 0.01664364 0.05072979 0.08700122 0.12471782 0.16328497 0.20223581 0.24121472 0.27996215 0.31830061 0.35612188 0.39337522 0.43005678 0.46619988 0.50186632 0.53713855 0.57211275 0.60689261 0.64158404 0.67629039 0.71110856 0.74612555 0.78141573 0.8170386 0.85303708 0.88943623 0.92624248 0.9634432 1.00100665 1.03888223 1.0770011 1.11527698 1.15360724

-1.47 -1.365 -1.26 -1.155 -1.05 -0.945 -0.84 -0.735 -0.63 -0.525 -0.42 -0.315 -0.21 -0.105 8.4655E-15 0.105 0.21 0.315 0.42 0.525 0.63 0.735 0.84 0.945 1.05 1.155 1.26 1.365 1.47 1.575 1.68 1.785 1.89 1.995 2.1 2.205 2.31 2.415 2.52 2.625 2.73 2.835 2.94 3.045 3.15 3.255 3.36

1.19187419 1.22994649 1.26768078 1.3049234 1.34151219 1.3772784 1.41204863 1.44564676 1.47789596 1.50862066 1.53764843 1.56481187 1.58995046 1.61291222 1.63355541 1.65174998 1.66737904 1.68034004 1.69054599 1.69792633 1.70242779 1.70401503 1.70267108 1.69839765 1.69121523 1.68116298 1.66829851 1.65269742 1.63445265 1.61367364 1.59048538 1.5650272 1.53745142 1.50792184 1.47661208 1.44370373 1.40938439 1.37384561 1.33728063 1.29988208 1.26183956 1.22333718 1.18455096 1.14564625 1.10677514 1.06807377 1.02965976

3.465 3.57 3.675 3.78 3.885 3.99 4.095 4.2 4.305 4.41 4.515 4.62 4.725 4.83 4.935 5.04 5.145 5.25 5.355 5.46 5.565 5.67 5.775 5.88 5.985 6.09 6.195 6.3

0.99162959 0.95405608 0.91698596 0.88043748 0.84439824 0.8088231 0.77363231 0.73870985 0.703902 0.66901619 0.6338202 0.5980416 0.56136764 0.52344557 0.48388329 0.44225063 0.39808105 0.35087393 0.30009744 0.24519212 0.18557509 0.12064498 0.04978767 -0.02761714 -0.11218855 -0.20453606 -0.30525034 -0.41489286

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