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
