IV.5.3. Les expressions de moment:

Inconnues (les angles de rotation)

Θ_B, Θ_C, Θ_D, Θ_E, Θ_F, Θ_G, Θ_H, Θ_I, Θ_J, Θ_K, Θ_L, Θ_M, Θ_O, Θ_P, Θ_Y, Θ_R, Θ_S, Θ_T, Θ_V, Θ_W, Θ_X, Θ_Y, Θ_Z, Θ_A’’, Θ_C’’, Θ_D’’, Θ_E’’, Θ_F’’, Θ_G’’, Θ_H’’, Θ_J’’, Θ_K’’, Θ_L’’, Θ_M’’, Θ_N’’, Θ_O’’, et ∆₁₁, ∆₂₂, ∆₃₃, ∆₄₄, ∆₅₅, ∆₆₆,

Connues au niveau de l’encastrement

∆_A= ∆_N= ∆_U=∆_B’’= ∆_I’’=∆_P’’=0

Θ_A= Θ_N= Θ_u= Θ_B’’ = Θ_I’’= Θ_P’’=0

Alors:

M_AB = (Θ_B- )

M_BA= (2Θ_B - )

M_NM = (Θ_M- )

M_UT = (Θ_T - )

M_TU = (2Θ_T- )

M_{B’’A’’} = (Θ_A’’- )

M_{A’’B’’} = (2Θ_A’’ - )

M_{I’’H’’} = (Θ_H’’ - )

M_{H’’I’’} = (2Θ_H’’- )

M_{P’’O’’} = (Θ_o’’- )

M_{O’’P’’} = (2Θ_O’’- )

M_BC = (2Θ_B+ Θ_C - )

M_CB = (2Θ_C+ Θ_B - )

M_ML = (2Θ_M+ Θ_L  - )

M_C = (2Θ_C+ Θ_B  - )

M_ML = (2Θ_M+ Θ_L - )

M_LM = (2Θ_L+ Θ_M - )

M_TS = (2Θ_T+ Θ_S - )

M_A’’Z = (2Θ_A’’+ Θ_Z - )

M_ZA’’= (2Θ_Z’’+ Θ_A’’ - )

M_{H’’G’’} = (2Θ_H’’+ Θ_G’’ - )

M_{G’’H’’} = (2Θ_G’’+ Θ_H’’ - )

M_{O’’N’’} = (2Θ_O’’+ Θ_N’’ - )

M_{N’’O’’} = (2Θ_N’’+ Θ_O’’  - )

M_CD = (2Θ_C+ Θ_D - )

M_DC = (2Θ_D+ Θ_C - )

M_LK = (2Θ_L+ Θ_M  - )

M_KL = (2Θ_K+ Θ_L  - )

M_SR = (2Θ_S+ Θ_R - )

M_RS = (2Θ_R+ Θ_S - )

M_ZY = (2Θ_Z+ Θ_Y - )

M_YZ = (2Θ_Y+ Θ_Z - )

M_{G’’F’’} = (2Θ_G’’+ Θ_F’’ - )

M_{N’’M’’}= (2Θ_N’’+ Θ_M’’  - )

M_DE= (2Θ_D+ Θ_E - )

M_ED = (2Θ_E+ Θ_D - )

M_KJ= (2Θ_K+ Θ_J - )

M_{F’’G’’} = (2Θ_F’’+ Θ_G’’ - )

M_{M’’N’’} = (2Θ_M’’+ Θ_N’’  - )

M_jK = (2Θ_j+ Θ_K  - )

M_Rq = (2Θ_R+ Θ_q - )

M_yx= (2Θ_y+ Θ_x - )

M_xy = (2Θ_F’’+ Θ_G’’ - )

M_{F’’E’’} = (2Θ_F’’+ Θ_E’’  - )

M_{E’’F’’} = (2Θ_E’’+ Θ_F’’  - )

M_{M’’L’’} = (2Θ_M’’+ Θ_L’’ - )

M_{L’’M’’} = (2Θ_L’’+ Θ_M’’ - )

M_EF= (2Θ_E+ Θ_F - )

M_FE = (2Θ_F+ Θ_E - )

M_JI = (2Θ_J+ Θ_I  - )

M_IJ = (2Θ_I+ Θ_J  - )

M_qP = (2Θ_q+ Θ_P - )

M_Pq = (2Θ_P+ Θ_q - )

M_xw= (2Θ_x+ Θ_w - )

M_wx = (2Θ_w+ Θ_x - )

M_{E’’D’’} = (2Θ_E’’+ Θ_D’’  - )

M_{D’’E’’} = (2Θ_D’’+ Θ_E’’  - )

M_{L’’K’’} = (2Θ_L’’+ Θ_K’’ - )

M_{K’’L’’} = (2Θ_K’’+ Θ_L’’ - )

M_FG= (2Θ_F+ Θ_G - )

M_GF = (2Θ_G+ Θ_F - )

M_IH = (2Θ_I+ Θ_H  - )

M_HI= (2Θ_H+ Θ_I  - )

M_PO = (2Θ_P+ Θ_O - )

M_OP = (2Θ_O+ Θ_P - )

M_WV= (2Θ_W+ Θ_V - )

M_VW= (2Θ_V+ Θ_W - )

M_{D’’C’’} = (2Θ_D’’+ Θ_C’’  - )

M_{C’’D’’}= (2Θ_C’’+ Θ_D’’  - )

M _{K’’J’’}= (2Θ_K’’+ Θ_J’’ - )

M_{J’’K’’}= (2Θ_J’’+ Θ_K’’ - )

M_WV= (2Θ_W+ Θ_V - )

M_VW= (2Θ_V+ Θ_W - )

M_{D’’C’’} = (2Θ_D’’+ Θ_C’’  - )

M_{C’’D’’}= (2Θ_C’’+ Θ_D’’  - )

M _{K’’J’’}= (2Θ_K’’+ Θ_J’’ - )

M_{J’’K’’}= (2Θ_J’’+ Θ_K’’ - )

M_BM= (2Θ_B+ Θ_M - )

M_MB= (2Θ_M+ Θ_B - )

M_{D’’C’’} = (2Θ_D’’+ Θ_C’’  - )

M_{C’’D’’}= (2Θ_C’’+ Θ_D’’  - )

M _{K’’J’’}= (2Θ_K’’+ Θ_J’’ - )

M _{J’’K’’}= (2Θ_J’’+ Θ_K’’ - )

M _BM= (2Θ_B+ Θ_M)-59.802

M _MB= (2Θ_M+ Θ_B)+59.802

M _MT= (2Θ_M+ Θ_T)-59.802

M _TM= (2Θ_T+ Θ_M)+59.802

M _TA’’= (2Θ_T+ Θ_A’’)-59.802

M _A’’T= (2Θ_A’’+ Θ_T)+59.802

M _{A’’H’’}= (2Θ_A’’+ Θ_H’’)-59.802

M _{H’’A’’}= (2Θ_H’’+ Θ_A’’)+59.802

M _{H’’O’’}= (2Θ_H’’+ Θ_O’’)-59.802

M _{O’’H’’}= (2Θ_O’’+ Θ_H’’)+59.802

M _CL= (2Θ_C+ Θ_L)-59.802

M _LC= (2Θ_L+ Θ_C)+59.802

M _LS= (2Θ_L+ Θ_S)-59.802

M _SL= (2Θ_S+ Θ_L)+59.802

M _SZ= (2Θ_S+ Θ_Z)-59.802

M _ZS= (2Θ_Z+ Θ_S)+59.802

M _ZG’’= (2Θ_Z+ Θ_G’’)-59.802

M _G’’Z= (2Θ_G’’+ Θ_Z)+59.802

M _{G’’N’’}= (2Θ_G’’+ Θ_N’’)-59.802

M _{N’’G’’}= (2Θ_N’’+ Θ_G’’)+59.802

M _DK= (2Θ_D+ Θ_K)-43.802

M _KD= (2Θ_K+ Θ_D)+43.802

M _KR= (2Θ_K+ Θ_R) -43.802

M _RK= (2Θ_R+ Θ_K)+43.802

M _RY= (2Θ_R+ Θ_Y) -43.802

M _YR= (2Θ_Y+ Θ_R)+43.802

M _YF’’= (2Θ_Y+ Θ_F’’) -43.802

M _F’’Y= (2Θ_F’’+ Θ_Y)+43.802

M _{F’’M’’}= (2Θ_F’’+ Θ_M’’) -43.802

M _{M’’F’’}= (2Θ_M’’+ Θ_F’’)+41.802

M _EJ= (2Θ_E+ Θ_J)-41.802

M _JE= (2Θ_J+ Θ_E) +41.802

M _JQ= (2Θ_J+ Θ_Q)-41.802

M _QJ= (2Θ_Q+ Θ_J) +41.802

M _QX= (2Θ_Q+ Θ_X)-41.802

M _XQ= (2Θ_X+ Θ_Q) +41.802

M_XE’’= (2Θ_X+ Θ_E’’)-41.802

M _E’’X= (2Θ_E’’+ Θ_X) +41.802

M _{E’’L’’}= (2Θ_E’’+ Θ_L’’)-41.802

M _{L’’E’’}= (2Θ_L’’+ Θ_E’’)+41.802

M _FI= (2Θ_F+ Θ_I) -41.802

M_IF= (2Θ_I+ Θ_F)+41.802

M _IP= (2Θ_I+ Θ_P) -41.802

M _PI= (2Θ_P+ Θ_I)+41.802

M _PW= (2Θ_P+ Θ_W)-41.802

M _WP= (2Θ_W+ Θ_P) +41.802

M_WD’’= (2Θ_W+ Θ_D’’)-41.802

M _D’’W= (2Θ_D’’+ Θ_W) +41.802

M _{D’’K’’}= (2Θ_D’’+ Θ_K’’)-41.802

M _{K’’D’’}= (2Θ_K’’+ Θ_D’’)+41.802

M _GH= (2Θ_G+ Θ_H) -21.802

M_HG= (2Θ_W+ Θ_D’’)+21.802

M _HO= (2Θ_H+ Θ_O) -21.802

M _OH= (2Θ_O+ Θ_H)+21.802

M _OV= (2Θ_O+ Θ_V) -21.802

M _VC’’= (2Θ_V+ Θ_C’’) +41.802

M_C’’V= (2Θ_C’’+ Θ_V)-41.802

M _D’’W= (2Θ_D’’+ Θ_W) +41.802

M _{D’’K’’}= (2Θ_D’’+ Θ_K’’)-41.802

M _{K’’D’’}= (2Θ_K’’+ Θ_D’’)+41.802

M _GH= (2Θ_G+ Θ_H) -21.46

M_HG= (2Θ_W+ Θ_D’’)+21.46

M _HO= (2Θ_H+ Θ_O) -21.46

M _OH= (2Θ_O+ Θ_H)+21.46

M _OV= (2Θ_O+ Θ_V) -21.46

M _VO= (2Θ_V+ Θ_O) +21.46

M _VC’’= (2Θ_V+ Θ_C’’)-21.46

M _C’’V= (2Θ_C’’+ Θ_v) +21.46

M _{C’’J’’}= (2Θ_C’’+ Θ_J’’)-21.46

M _{J’’C’’}= (2Θ_J’’+ Θ_C’’) +21.46

Tableau de resultants (angles de rotations et replacement)

ΘB	33.35496676	ΘK	9.332302629		ΘT	17.00228744		ΘE’’	5.68908294
ΘC	29.37011195	ΘL	-2.157173579		ΘV	1.004860253		ΘF’’	11.99824404
ΘD	23.73794097	ΘO’’	-2.26030742		ΘW	2.924584428		ΘG’’	10.4265224
ΘE	2.696173319	ΘM	8.839975511		ΘX	5.405710167		ΘH’’	11.43495447
ΘF	15.24813034	ΘO	1.200415074		ΘY	8.500936198		ΘJ’’	-3.777855215
ΘG	7.775911019	ΘP	2.890984755		ΘZ	12.76380038		ΘK’’	-3.425694128
ΘH	0.591142368	Θq	6.779799316		ΘA’’	10.6225577		ΘL’’	-0.955921674
ΘI	1.638341092	ΘR	11.72399603		ΘC’’	2.121391834		ΘM’’	-0.230151732
ΘJ	5.857827341	ΘS	12.72916111		ΘD’’	3.466146696		ΘN’’	22.09983635
∆11=79.03147815				∆22=89.46109756			∆33=76.77703128
∆44=49.53096595				∆55=27.98064484			∆66=13.87282771

En remplaçant les angles des rotations et les déplacements sur les expressions des moments, nous trouvons alors chaque valeur de ces derniers.

MOMENTS FLECHISSANTS(KNm)

 

M_IH=-2.97

M_AB=-19.73 KNm

M_HI=3.99

M_BA=1.6 KNm

M_BC=20.26

M_CB=-14.51 KNm

M_CD=-9.30 KNm

M_DC=-9.30 KNm

M_DE=12.14 KNm

M_ED=-9.73 KNm

M_EF=15.06 KNm

M_FE=-12.71 KNm

M_GF=17.17KNm

M_FG=-14.57 KNm

M_NM=-31.80 KNm

M_MN=25.72KNm

M_ML=-22.86 KNm

M_LM=23.66 KNm

M_LK=-16.23 KNm

M_KL=17.92KNm

M_KJ=-11.45KNm

M_JK=-13.31 KNm

M_JI=-6.90KNm

M_IJ=8.74 KNm

M_UT=-30.62 KNm

M_TU=23.35 KNm

M_TS=--18.95KNm

M_ST=20.71KNm

M_SR=-14.56 KNm

M_RS=-16.35KNm

M_RQ=10.35 KNm

M_QR=12.27 KNm

M_QP=-5.98 KNm

M_PQ=7.90KNm

M_PO=-1.81KNm

M_OP=-1.81 KNm

M_PO=3.25KNm

M_{B’’A’’}=-31.83 KNm

M_{A’’B’’}=23.74KNm

M_A’’Z=-19.42 KNm

M_ZA’’=20.99KNm

M_ZY=-14.72KNm

M_YZ=16.60KNm

M_YX=-10.41KNm

M_XY=12.34 KNm

M_xw=-6.03KNm

M_wx=7.95 KNm

M_wv=-1.85KNm

M_vw=3.30KNm

M_{I’’H’’}=-30.35KNm

M_{H’’I’’}=-10.41KNm

M_{G’’F’’}=-17.67 KNm

M_{F’’G’’}=19.55KNm

M_{F’’E’’}=10.10 KNm

M_{E’’F’’}=11.96KNm

M_{E’’D’’}= -5.52KNm

M_{D’’E’’}= 7.38 KNm

M_{D’’C’’}=-1.19KNm

M_{C’’D’’}=2.62 KNm

M_{J’’K’’}=-18.42KNm

M_{K’’J’’}=-17.56KNm

M_{K’’L’’}=23.50 KNm

M_{L’’M’’}= 25.58KNm

M_{M’’L’’}= -22.93 KNm

M_{M’’N’’}=30.49KNm

M_{N’’M’’}=30.70 KNm

M_{N’’O’’}=-39.59KNm

M_{O’’N’’}=38.69KNm

M_{P’’O’’}=33.81KNm

M_{O’’P’’}= 35.69KNm

M_BM=-88.89KNm

M_MB=-18.66KNm

M_MT=-40.33 KNm

M_TM=-80.48KNm

M_IP=-46.63 KNm

M_PI=-37.17KNm

M_PW= -46.67KNm

M_WP=-36.92 KNm

M_WP=-46.88KNm

M_WD’’=-46.88 KNm

M_D’’W=-36.87KNm

M_{D’’K’’}=-41.05KNm

M_{K’’D’’}=-38.31KNm

M_TA’’= -81.24KNm

M_A’’T= -38.18KNm

M_{A’’H’’}=-82.70KNm

M_{H’’A’’}=-38.08KNm

M_{H’’O’’}=-72.50KNm

M_{O’’H’’}=-41.59KNm

M_CL=-14.51KNm

M_LC= -82.70 KNm

M_LS=-42.81 KNm

M_SL=-77.27KNm

M_SZ=-41.81 KNm

M_ZS=-77.57KNm

M_GH=-21.00KNm

M_HG=-11.57KNm

M_HO= 20.02KNm

M_OH= -23.07KNm

M_{O’’V’’}=-19.83KNm

M_{V’’O’’}=-23.07KNm

M_ZG’’=-41.86KNm

M_G’’Z=-78.01KNm

M_{G’’N’’}=-44.52KNm

M_{N’’G’’}= -70.30 KNm

M_DK= -60.13KNm

M_{O’’V’’}=-21.44KNm

M_KR=-57.14KNm

M_RK=-30.14KNm

M_RY=-57.31KNm

M_YR=-30.29KNm

M_YF’’= -57.36KNm

M_F’’Y= -30.30KNm

M_VC’’=-53.36KNm

M_C’’V=-31.18KNm

M_{C’’J’’}=-20.69 KNm

M_{J’’C’’}=-18.42KNm

M_{F’’M’’}=-31.18 KNm

M_{M’’F’’}=-53.42KNm

M_EJ= -24.79KNm

M_JE=--53.20KNm

M_JQ=-32.98KNm

M_QJ=-50.87 KNm

M_QX=-32.62KNm

M_XQ=-50.87KNm

M_XE’’=-32.60KNm

M_E’’X= -51.12KNm

M_{E’’L’’}= -33.65KNm

M_{L’’E’’}=-46.35KNm

M_FI=-48.48KNm

M_IF=-29.88KNm

EFFORTS TRANCHANTS (KN)

T_BM= -11.59KNm

T_MB=-79.66KNm

T_MT=-79.66KNm

T_TM=-78.94KNm

T_TA’’=-78.72KNm

T_A’’T=-78.72KNm

T_{A’’H’’}= - 78.72 KNm

T_{H’’A’’}= -81.98  KNm

T_{H’’O’’}= -81.98KNm

T_{O’’H’’}=-97.43KNm

T_CL=-7.78 KNm

T_LC=-13.24KNm

T_LS=-103.24KNm

T_SL=-98.32 KNm

T_SZ= -98.32 KNm

T_ZS= -98.60 KNm

T_ZG’’= 98.60 KNm

T_G’’Z=-83.26 KNm

T_{G’’N’’}=-83.26KNm

T_{N’’G’’}=-97.43KNm

T_DK =-7.29KNm

T_KD=-59.11KNm

T_KR= -59.95KNm

T_RY= -58.95KNm

T_YR= -58.94 KNm

T_YF’’=-58.94 KNm

T_F’’Y=-72.47KNm

T_{F’’M’’}=-72.47KNm

T_{M’’F’’}=-71.26KNm

T_EJ=-9.26KNm

T_JE= -69.81KNm

T_JQ= -67.29KNm

T_QJ= -67.18 KNm

T_QX=-67.18 KNm

T_XQ=-67.29KNm

T_XE’’=67.29KNm

T_E’’X=-59.53KNm

T_{E’’L’’}=59.53KNm

T_{L’’E’’}= -16.17 KNm

T_BM= -11.59KNm

T_MB=-79.66KNm

T_MT=-79.66KNm

T_TM=-78.94KNm

T_TA’’=-78.72KNm

T_A’’T=-78.72KNm

T_{A’’H’’}= - 78.72 KNm

T_{H’’A’’}= -81.98  KNm

T_{H’’O’’}= -81.98KNm

T_{O’’H’’}=-97.43KNm

T_CL=-7.78 KNm

T_LC=-13.24KNm

T_LS=-103.24KNm

T_SL=-98.32 KNm

T_SZ= -98.32 KNm

T_ZS= -98.60 KNm

T_FI= -10.58 KNm

T_IF=-60.34 KNm

T_IP=-60.34KNm

T_PI=-60.34KNm

T_PW=-60.27KNm

T_WP=--60.20KNm

T_WD’’= -60.20KNm

T_D’’W= -1.27KNm

T_{D’’K’’}= 1.27KNm

T_{K’’D’’}=1.27 KNm

T_GH=28.93KNm

T_HG=31.95KNm

T_H0=-34.55KNm

T_OH=-32.95KNm

T_OV= -32.95KNm

T_VO= -33.00KNm

T_VC’’= -33.00 KNm

T_C’’V=-33.07KNm

T_{C’’J’’}=--33.07KNm

T_{J’’C’’}=-31.62KNm

Comme nous venons de trouver les moments en travée et aux appuis de chaque barre de notre portique, nous allons considérer les efforts maximaux pour chaque niveau à l’aide de quoi nous trouverons les sections d’armatures dans chacun de cas cité ci-haut (poutre longitudinale).