Navier108 - Calcul d'une dalle rectangulaire - Solution de
Navier avec les sries de Fourrierjuin 2012xyarticule sur ses 4
ctsH. Thoniery(pour excuter, taper la touche F9 aprs la fin des
donnes)L'auteur n'est pasflche // Ox00000dalle et chargespoints de
calculM et V maxiLmin/LmaxM/(p.L2)Eh3f/(pL4)responsable
de000.50.099980.11668l'usage fait de010Epaisseur dalleh1mce
programme2002010Hauteur utiled0.9m2010200Longueur
dalleA20m01000Largeur dalleB10m00Module d'YoungE10,000MPa000Coeff.
de Poissonn0.21002001020002010201010200Charges rparties sur des
rectangles de cts a2 x b2010100a1a2b1b2p00mmmmkN/m2Charg. kNAire
m200010200101200200.00200000200.0000000300.0000000400.0000000500.0000600.00Flche
// Oy000200200.0030000000000RsultatsM i l i e u x d e s b o r d
sMilieuVos coordonnes00000gauchedroithautbaspanneaude
calcul00000Abscisse xm0201010101010000Ordonne ym551005505000Moment
MakNm/m005.29083471951322E-1603.68005287083.68005287080000400000Moment
MbkNm/m00009.99836370769.9983637076000000000Eff. tranchant
VakN/m3.4111329937-3.4111329937-1.00105854832329E-310-1.44941625100671E-16-1.44941625100671E-1603.41113299370000000Eff.
tranchant
VbkN/m0-6.8005875056913E-32-4.52011213424.52011213427.7147641891796E-177.7147641891796E-174.5201121342000Aciers
// 0x : moment max () =3.8073222287kNm/m00000Moment
rduitMPa02.05943702913594E-181.97778481920226E-1800.01234365890.01234365890000Aciers
// 0y: moment max ()
=9.9983637076kNm/m00CisaillementMPa0.00379014780.00379014780.00502234680.00502234681.61046250111857E-191.61046250111857E-190.00502234680.003790147800Effort
tranchant // Ox (x)
=4.5201121342kN/m000Flchemm0-1.7748620099684E-17-1.46318929439888E-170-0.1166844181-0.11668441810000Effort
tranchant // Oy (x) =3.4111329937kN/m50000000000Nota : calculs
longs, patientezMomentM.rduitEff. tr.maxCisaillementDforme00000Sur
les bords, les rsultats de convergence des sries sont
mdiocresmaximalmaximalmaximalmaximalmaximale00000Cisaillement v =
V/d9.99836370760.0123436589<
3,24.52011213420.00590864330.117mm00Moment rduit : b = M /
d2MNm/mMPaMN/mMPa000Pour les plages indiques ci-dessous et des
coordonnes ci-dessus600000Plage :x =0y =0Ma max xet ycx00000de
variation x =20 y =103.8073222287650.0300000kNm/mcy00000Moment
Ma0123456789100.0300kNm/mx =02468101214161820x =Momenty
=0000000000000A mi-porte //
OxMa5.975101.08042272491.30546425511.3037992791.27712911771.27022097821.27712911771.3037992791.30546425511.08042272496.7737908218427E-1613.807322228765.03105201.89795612782.35483002272.37016810232.31711394872.29772796092.31711394872.37016810232.35483002271.897956127802x6.035100302.45930479043.11961277973.15881433933.08599414183.05577959293.08599414183.15881433933.11961277972.459304790403664.9705402.78713901523.58434061523.64357183313.55926117823.52225249883.55926117823.64357183313.58434061522.787139015204y5.97500502.89519971093.74035657533.80732222873.71932729743.68005287083.71932729743.80732222873.74035657532.895199710905500602.78713901523.58434061523.64357183313.55926117823.52225249883.55926117823.64357183313.58434061522.78713901520665702.45930479043.11961277973.15881433933.08599414183.05577959293.08599414183.15881433933.11961277972.459304790407Mb9.975105801.89795612782.35483002272.37016810232.31711394872.29772796092.31711394872.37016810232.35483002271.8979561278089.9983637076105.0305901.08042272491.30546425511.3037992791.27712911771.27022097821.27712911771.3037992791.30546425511.08042272496.7737908218427E-169x10.0351001004.58372651388668E-165.44152545468818E-165.39581181063845E-165.29963220477637E-165.29083471951322E-165.29963220477637E-165.39581181063845E-165.44152545468818E-164.58372651388668E-163.0495383002771E-311010104.97y9.975Mx(1/2)My(1/2)0000522.895199710913.7323023421Ma
max xet
y43.740356575326.53093408489.9983637076105Va-0.034.9763.807322228738.4716614658kNm/m3.41113299370.035.0383.719327297449.6179168105Moment
Mb012345678910x103.680052870859.9983637076kNm/mx
=02468101214161820x =Moment0-0.035.03123.719327297469.6179168105y
=0000000000000A mi-porte //
Oyy0.034.97143.807322228778.4716614658101.70377601692.70907170053.30542734113.62842244243.73230234213.62842244243.30542734112.70907170051.70377601698.06511531046249E-1615163.740356575386.5309340848202.76659855684.60203717565.72819288836.33745327446.53093408486.33745327445.72819288834.60203717562.766598556802182.895199710993.7323023421303.42415388245.84468263867.37413272868.20737773098.47166146588.20737773097.37413272865.84468263863.424153882403Vb9.97-0.03200100403.79282433976.55486611368.33307392789.30822729959.61791681059.30822729958.33307392786.55486611363.7928243397044.520112134210.030.03503.9136049266.78747308518.64932301769.67300190299.99836370769.67300190298.64932301766.78747308513.91360492605x603.79282433976.55486611368.33307392789.30822729959.61791681059.30822729958.33307392786.55486611363.792824339706109.970.03703.42415388245.84468263867.37413272868.20737773098.47166146588.20737773097.37413272865.84468263863.424153882407y10.03-0.03802.76659855684.60203717565.72819288836.33745327446.53093408486.33745327445.72819288834.60203717562.7665985568080901.70377601692.70907170053.30542734113.62842244243.73230234213.62842244243.30542734112.70907170051.70377601698.0651153104625E-1691007.90588552943259E-1600000007.90588552943259E-164.11723920617501E-3110Vx(1/2)Va
max xet y03.41113299373.411132993705Eff. Tranchant22.035212231kN/mA
mi-porte // Ox41.1392094755Eff. Tranch. Vax
=01234567891060.5756355387kN/m0246810121416182080.2133913017y
=00000000000010-1.44941625100671E-1611.53361015390.68154842130.37810534010.18748038490.0538942383-1.45371630908911E-16-0.0538942383-0.1874803849-0.3781053401-0.6815484213-1.533610153912-0.213391301722.43904863311.27778524390.69521183020.34593494830.1174394401-1.5765753533022E-16-0.1174394401-0.3459349483-0.6952118302-1.2777852439-2.439048633114-0.575635538732.99378668511.70368985280.93696248940.46959232410.1693323734-1.49769914284335E-16-0.1693323734-0.4695923241-0.9369624894-1.7036898528-2.993786685116-1.139209475543.30729339511.95348498771.08790620040.54848003070.2022494787-1.45388067718868E-16-0.2022494787-0.5484800307-1.0879062004-1.9534849877-3.307293395118-2.03521223153.41113299372.0352122311.13920947550.57563553870.2133913017-1.44941625100671E-16-0.2133913017-0.5756355387-1.1392094755-2.035212231-3.411132993720-3.411132993763.30729339511.95348498771.08790620040.54848003070.2022494787-1.45388067718868E-16-0.2022494787-0.5484800307-1.0879062004-1.9534849877-3.307293395172.99378668511.70368985280.93696248940.46959232410.1693323734-1.49769914284335E-16-0.1693323734-0.4695923241-0.9369624894-1.7036898528-2.993786685182.43904863311.27778524390.69521183020.34593494830.1174394401-1.5765753533022E-16-0.1174394401-0.3459349483-0.6952118302-1.2777852439-2.4390486331Vy(1/2)91.53361015390.68154842130.37810534010.18748038490.0538942383-1.45371630908911E-16-0.0538942383-0.1874803849-0.3781053401-0.6815484213-1.533610153904.5201121342107.35785851341085E-162.59313910484363E-161.56925661791086E-167.83852547307959E-171.54381917209718E-17-1.00105854832329E-31-1.54381917209718E-17-7.83852547307957E-17-1.56925661791086E-16-2.59313910484363E-16-7.35785851341085E-1613.645754796222.729992531231.801125223340.8813418331Eff.
Tranchant57.7147641891796E-17Vb max xet yA mi-porte //
Oy6-0.88134183314.52011213421007-1.8011252233kN/m8-2.7299925312Eff.
Tranch. Vb0123456789109-3.6457547962kN/mx
=0246810121416182010-4.5201121342y
=002.63352869283.68256750684.17468425174.43043495374.52011213424.43043495374.17468425173.68256750682.63352869280101.87466977292.85147546523.34315029983.57630255113.64575479623.57630255113.34315029982.85147546521.87466977298.87444858665042E-16201.28420281122.06342033442.47478566892.67151996132.72999253122.67151996132.47478566892.06342033441.28420281125.7157925776591E-16300.79407909731.32412634921.61725329151.75903820391.80112522331.75903820391.61725329151.32412634920.79407909733.40254897462404E-16400.36686891580.63336499550.78616685820.85992421310.88134183310.85992421310.78616685820.63336499550.36686891581.46412820842374E-1650-2.59279950621823E-172.76310577198337E-176.44836394816078E-177.67852496147435E-177.7147641891796E-177.67852496147435E-176.44836394816078E-172.76310577198337E-17-2.59279950621822E-17-6.8005875056913E-32fx(1/2)60-0.3668689158-0.6333649955-0.7861668582-0.8599242131-0.8813418331-0.8599242131-0.7861668582-0.6333649955-0.3668689158-1.46412820842374E-160070-0.7940790973-1.3241263492-1.6172532915-1.7590382039-1.8011252233-1.7590382039-1.6172532915-1.3241263492-0.7940790973-3.40254897462404E-162-0.043268817680-1.2842028112-2.0634203344-2.4747856689-2.6715199613-2.7299925312-2.6715199613-2.4747856689-2.0634203344-1.2842028112-5.7157925776591E-164-0.077180910290-1.8746697729-2.8514754652-3.3431502998-3.5763025511-3.6457547962-3.5763025511-3.3431502998-2.8514754652-1.8746697729-8.87444858665042E-166-0.0999084199100-2.6335286928-3.6825675068-4.1746842517-4.4304349537-4.5201121342-4.4304349537-4.1746842517-3.6825675068-2.6335286928-08-0.112623088310-0.116684418112-0.1126230883w
max xet
y14-0.09990841990.116684418110516-0.0771809102Dforme01234567891018-0.0432688176mmx
=0246810121416182020-1.7748620099684E-17y
=00000000000010-0.0139000053-0.0245403883-0.0315986804-0.0355360707-0.0367934775-0.0355360707-0.0315986804-0.0245403883-0.0139000053-5.7666432593128E-18fy(1/2)20-0.0260766356-0.0462234797-0.0596312914-0.0671157579-0.0695055265-0.0671157579-0.0596312914-0.0462234797-0.0260766356-1.07629629736489E-170030-0.035426115-0.0630107561-0.0814331399-0.0917273007-0.0950146219-0.0917273007-0.0814331399-0.0630107561-0.035426115-1.45697256677628E-171-0.036793477540-0.0412788648-0.0735782235-0.0952048196-0.1072996838-0.1111627357-0.1072996838-0.0952048196-0.0735782235-0.0412788648-1.69430513318933E-172-0.069505526550-0.0432688176-0.0771809102-0.0999084199-0.1126230883-0.1166844181-0.1126230883-0.0999084199-0.0771809102-0.0432688176-1.7748620099684E-173-0.095014621960-0.0412788648-0.0735782235-0.0952048196-0.1072996838-0.1111627357-0.1072996838-0.0952048196-0.0735782235-0.0412788648-1.69430513318933E-174-0.111162735770-0.035426115-0.0630107561-0.0814331399-0.0917273007-0.0950146219-0.0917273007-0.0814331399-0.0630107561-0.035426115-1.45697256677628E-175-0.116684418180-0.0260766356-0.0462234797-0.0596312914-0.0671157579-0.0695055265-0.0671157579-0.0596312914-0.0462234797-0.0260766356-1.07629629736489E-176-0.111162735790-0.0139000053-0.0245403883-0.0315986804-0.0355360707-0.0367934775-0.0355360707-0.0315986804-0.0245403883-0.0139000053-5.76664325931281E-187-0.0950146219100-5.54976369529031E-18-9.77412321332195E-18-1.25723188352649E-17-1.41332082105104E-17-1.46318929439888E-17-1.41332082105104E-17-1.25723188352649E-17-9.77412321332195E-18-5.54976369529031E-18-2.31170943764239E-338-0.06950552659-0.036793477510-1.46318929439888E-17
ABa1a2b1b2
Navier
012345678910
012345678910
012345678910
012345678910
012345678910
dalle et chargespoints de calculM et V maxi
Mx(1/2)
My(1/2)
Vy(1/2)
Vx(1/2)
fx(1/2)
fy(1/2)
pour des armatures // au ct Apour des armatures // au ct B
Attribute VB_Name = "Feuil1"Attribute VB_Base =
"0{00020820-0000-0000-C000-000000000046}"Attribute
VB_GlobalNameSpace = FalseAttribute VB_Creatable = FalseAttribute
VB_PredeclaredId = TrueAttribute VB_Exposed = TrueAttribute
VB_TemplateDerived = FalseAttribute VB_Customizable = True
Attribute VB_Name = "Module1"Function maximv(ta, code)Ma = 0For
i = 1 To 11Y = ta(i + 1, 1) For j = 1 To 11 X = ta(1, j + 1) If
Abs(ta(i + 1, j + 1)) > Ma Then Ma = Abs(ta(i + 1, j + 1)): Xi =
X: Yi = Y Next jNext iOn code GoTo 11, 12, 1311 maximv = Ma: GoTo
1412 maximv = Xi: GoTo 1413 maximv = Yi: GoTo 1414 End
FunctionFunction Macro4(H, E, NU, LA, LB, P0, A1, A2, B1, B2, X, Y,
KOD)'DATA 0.2, 12000, 0, 20, 5, 1, 5, 3, 3.2, 4, 4.2Rem : "NAVIER"
- Henry THONIER - Decembre 1988 -Rem: Calcul des Dalles sur 4
Appuis Articules avec Charges Partielles.Rem: Charges Constantes
sur Un ou Plusieurs Rectangles'"Epaisseur h (m)="; H'"Module
d'Young(MPa) ="; E'"Coefficient de Poisson="; NU'"Longueur Suivant
Ox(m)="; LA'"Longueur Suivant Oy(m)="; LB'"Nombre de Rectangles
Charges="; NR'"Charge N ## en MN/m2 ="; P0'"Abscisse Debut de
Charge(m)="; A1'"Abscisse Fin de Charge(m) ="; A2'"Ordonnee Debut
de Charge(m)="; B1'"Ordonnee Fin de Charge(m) ="; B2' X et Y
coordonnes du point calculDim A(31, 31), B(31, 31) As Double ' ,
Z(30)Dim Pi, D, U1, U2, U3, U4, U5, U6, U7, X1, Y1, C1, C2, T1, T2
As DoubleDim cos3, cos4, cos5, cos6, con1, com1, sin1, sim1 As
DoubleDim M, N, M1, N1 As IntegerPi = 4 * Atn(1) '3.141593A2 = A1 +
A2B2 = B1 + B2M1 = 14N1 = M1If P0 = 0 Then Ma = 0: MB = 0: VA = 0:
VB = 0: WA = 0: GoTo 1400For M = 1 To M1 For N = 1 To N1 A(M, N) =
0 Next NNext MD = E * H ^ 3 / 12 / (1 - NU * NU)U3 = A1 / LA * PiU5
= A2 / LA * PiU4 = B1 / LB * PiU6 = B2 / LB * PiZM = 0: ZV = 0 For
M = 1 To M1 For N = 1 To N1 cos3 = Cos(M * U3) cos5 = Cos(M * U5)
cos4 = Cos(N * U4) cos6 = Cos(N * U6) A(M, N) = 4 * P0 / M / N / Pi
/ Pi * (cos3 - cos5) * (cos4 - cos6) + A(M, N) U1 = (M * M / LA /
LA + N * N / LB / LB) * (M * M / LA / LA + N * N / LB / LB) B(M, N)
= -1 / D / Pi / Pi / Pi / Pi * A(M, N) / U1 Next N Next MC1 = 0C2 =
0WA = 0T1 = 0T2 = 0 For M = 1 To M1 For N = 1 To N1 X1 = X * Pi /
LA Y1 = Y * Pi / LB con1 = Cos(N * Y1) com1 = Cos(M * X1) sin1 =
Sin(N * Y1) sim1 = Sin(M * X1) U2 = sim1 * sin1 WA = WA + B(M, N) *
U2 U7 = (M * M / LA / LA + N * N / LB / LB) U1 = U7 * U7 C1 = C1 +
A(M, N) * (M * M / LA / LA + NU * N * N / LB / LB) * U2 / U1 C2 =
C2 + A(M, N) * (N * N / LB / LB + NU * M * M / LA / LA) * U2 / U1
T1 = T1 + A(M, N) * M / LA / U7 * com1 * sin1 T2 = T2 + A(M, N) * N
/ LB / U7 * sim1 * con1 Next N NextMa = C1 / Pi / Pi ' KOD=1MB = C2
/ Pi / Pi ' KOD=2VA = T1 / Pi ' KOD=3VB = T2 / Pi ' KOD=41400 If
KOD = 1 Then Macro4 = MaIf KOD = 2 Then Macro4 = MBIf KOD = 3 Then
Macro4 = VAIf KOD = 4 Then Macro4 = VBIf KOD = 5 Then Macro4 = WA99
End Function
Attribute VB_Name = "Module2"
Attribute VB_Name = "ThisWorkbook"Attribute VB_Base =
"0{00020819-0000-0000-C000-000000000046}"Attribute
VB_GlobalNameSpace = FalseAttribute VB_Creatable = FalseAttribute
VB_PredeclaredId = TrueAttribute VB_Exposed = TrueAttribute
VB_TemplateDerived = FalseAttribute VB_Customizable = True
