metodo cros
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MÉTODO CROSS
W=3.797 W=3.92 W=3.873 W=3.92H4=9.493Tn
T U V W X
W=3.797 W=3.92 W=3.873 W=3.92H3=16.612Tn
O P Q R S
W=3.797 W=3.92 W=3.873 W=3.92H2=21.358Tn
K L M N Ñ
W=3.797 W=3.92 W=3.873 W=3.92H1=23.732Tn
F G H I J
A B C D E
6.00 4.50 5.00 4.50
ELEMENTOS ESTRUCTURALES MODULO DE ELASTICIDAD DEL CONCRETO
ELEMENTO SECCIÓN INERCIA E = 234787.137637COLUMNA: 0.35 X 0.35 0.001251
VIGA P : 0.25 X 0.50 0.002604VIGA S : 0.25 X 0.40 0.001333
CARGAS
CARGAS HORIZONTALESH1 = 23.732 TnH2 = 21.358 TnH3 = 16.612 TnH4 = 9.493 Tn
PESO DE CARGAS DISTRIBUIDAS ( W )TRAMO 1 TRAMO2 TRAMO3 TRAMO4
WFG = 3.797 Tn/m WGH = 3.920 Tn/m WHI = 3.873 Tn/m WIJ = 3.920WKL = 3.797 Tn/m WLM = 3.920 Tn/m WMN = 3.873 Tn/m WNÑ = 3.920WOP = 3.797 Tn/m WPQ = 3.920 Tn/m WQR = 3.873 Tn/m WRS = 3.920WTU = 3.797 Tn/m WUV = 3.920 Tn/m WVW = 3.873 Tn/m WWX = 3.920
LONGITUDES
LONGITUD DE VIGASFG = KL = OP = TU = 6GH = LM = PQ = UV = 4.5HI = MN = QR = VW = 5IJ = NÑ = RS = WX = 4.5
LONGITUD DE COLUMNASAF = BG = CH = DI = JE = 2.9FK = GL = HM = IN = JÑ = 2.9KO = LP = MQ = NR = ÑS = 2.9TO = PU = QV = RW = SX = 2.9
RIGIDECES RELATIVAS
RIGIDECES RELATIVAS K = I/LC KAF = 4.312E-04 C KFK = 4.312E-04 C KKO = 4.312E-04 C KOTC KBG = 4.312E-04 C KGL = 4.312E-04 C KLP = 4.312E-04 C KPUC KCH = 4.312E-04 C KHM = 4.312E-04 C KMQ = 4.312E-04 C KQVC KDI = 4.312E-04 C KIN = 4.312E-04 C KNR = 4.312E-04 C KRWC KEJ = 4.312E-04 C KJÑ = 4.312E-04 C KÑS = 4.312E-04 C KSX
V KFG = 4.340E-04 V KKL = 4.340E-04 V KOP = 4.340E-04 V KTUV KGH = 5.787E-04 V KLM = 5.787E-04 V KPQ = 5.787E-04 V KUVV KHI = 5.208E-04 V KMN = 5.208E-04 V KQR = 5.208E-04 V KVWV KIJ = 5.787E-04 V KNÑ = 5.787E-04 V KRS = 5.787E-04 V KWX
MᵒFG = - MᵒGF = - 11.391 MᵒGH = - MᵒHG = - 6.615MᵒKL = - MᵒLK = - 11.391 MᵒLM = - MᵒML = - 6.615MᵒOP = - MᵒPO = - 11.391 MᵒPQ = - MᵒQP = - 6.615MᵒTU = - MᵒUT = - 11.391 MᵒUV = - MᵒVU = - 6.615
MOMENTOS DE EMPOTRAMIENTO ( Mᵒ )
MᵒHI = - MᵒIH = - 8.069 MᵒIJ = - MᵒJI = - 6.615MᵒMN = - MᵒNM = - 8.069 MᵒNÑ = - MᵒÑN = - 6.615MᵒQR = - MᵒRQ = - 8.069 MᵒRS = - MᵒSR = - 6.615MᵒVW = - MᵒWV = - 8.069 MᵒWX = - MᵒXW = - 6.615
CROSS SIN DESPLAZAMIENTO
MOMENTOS DE EMPOTRAMIENTOS
MᵒFG = - MᵒGF = - 11.391MᵒKL = - MᵒLK = - 11.391MᵒOP = - MᵒPO = - 11.391MᵒTU = - MᵒUT = - 11.391
MᵒGH = - MᵒHG = - 6.615MᵒLM = - MᵒML = - 6.615MᵒPQ = - MᵒQP = - 6.615MᵒUV = - MᵒVU = - 6.615
MᵒHI = - MᵒIH = - 8.069MᵒMN = - MᵒNM = - 8.069MᵒQR = - MᵒRQ = - 8.069MᵒVW = - MᵒWV = - 8.069
MᵒIJ = - MᵒJI = - 6.615MᵒNÑ = - MᵒÑN = - 6.615MᵒRS = - MᵒSR = - 6.615MᵒWX = - MᵒXW = - 6.615
FACTORES DE DISTRIBUCIÓN
1ER NIVEL 1ER NIVEL 2DO NIVEL 3ER NIVEL
NUDO NUDO NUDO NUDO
A AF = 0.000F
FA = 0.333K
KF = 0.333O
OK = 0.333B BG = 0.000 FK = 0.333 KL = 0.335 OP = 0.335C CH = 0.000 FG = 0.334 KO = 0.332 OT = 0.332D DI = 0.000
G
GB = 0.230
L
LG = 0.230
P
PL = 0.230E EJ = 0.000 GF = 0.231 LK = 0.231 PO = 0.231
GL = 0.230 LP = 0.230 PU = 0.230GH = 0.309 LM = 0.309 PQ = 0.309
H
HC = 0.220
M
MH = 0.220
Q
QM = 0.220HG = 0.295 ML = 0.295 QP = 0.295HM = 0.220 MQ = 0.220 QV = 0.220
FACTORES DE DISTRIBUCION
FACTORES DE DISTRIBUCION
FACTORES DE DISTRIBUCION
FACTORES DE DISTRIBUCION
H
HI = 0.265
M
MN = 0.265
Q
QR = 0.265
I
ID = 0.220
N
NI = 0.220
R
RN = 0.220IH = 0.265 NM = 0.265 RQ = 0.265IN = 0.220 NR = 0.220 RW = 0.220IJ = 0.295 NÑ = 0.295 RS = 0.295
JJE = 0.299
ÑÑJ = 0.299
SSÑ = 0.299
JI = 0.402 ÑN = 0.402 SR = 0.402JÑ = 0.299 ÑS = 0.299 SX = 0.299
TO TU UT UP UV0.498 0.502 0.301 0.299 0.400
-11.39 11.39 -6.615.67 5.72 -1.44 -1.43 -1.911.89 -0.72 2.86 -0.55 0.27-0.58 -0.59 -0.78 -0.77 -1.03-0.69 -0.39 -0.29 -0.13 0.200.54 0.54 0.07 0.07 0.090.16 0.03 0.27 0.12 0.03-0.10 -0.10 -0.13 -0.13 -0.17
0.00 6.89 -6.89 0.00 11.95 -2.82 -9.14
OK OT OP PO PL PU PQ0.333 0.332 0.335 0.231 0.230 0.230 0.309
-11.39 11.39 -6.613.79 3.78 3.82 -1.10 -1.10 -1.10 -1.481.89 2.84 -0.55 1.91 -0.27 -0.71 0.21-1.39 -1.39 -1.40 -0.26 -0.26 -0.26 -0.35-0.54 -0.29 -0.13 -0.70 -0.07 -0.39 0.080.32 0.32 0.32 0.25 0.25 0.25 0.330.17 0.27 0.12 0.16 0.06 0.03 -0.04-0.19 -0.19 -0.19 -0.05 -0.05 -0.05 -0.07
0.00 4.06 5.34 -9.40 0.00 11.59 -1.44 -2.23 -7.92
KO KF KL LK LP LG LM0.332 0.333 0.335 1 0.231 0.230 0.230 0.309 1
-11.39 11.39 -6.613.78 3.79 3.82 -1.10 -1.10 -1.10 -1.481.90 1.90 -0.55 1.91 -0.55 -0.55 0.21-1.08 -1.08 -1.09 -0.24 -0.24 -0.24 -0.32-0.70 -0.22 -0.12 -0.54 -0.13 -0.18 0.09
0.34 0.35 0.35 0.18 0.18 0.18 0.240.16 0.12 0.09 0.17 0.12 0.03 -0.03-0.12 -0.12 -0.12 -0.07 -0.07 -0.07 -0.09
0.00 4.29 4.73 -9.02 0.00 11.70 -1.78 -1.93 -7.99
FA FK FG GF GB GL GH0.333 0.333 0.334 1 0.231 0.230 0.230 0.309 1
-11.39 11.39 -6.613.79 3.79 3.80 -1.10 -1.10 -1.10 -1.48
1.90 -0.55 1.90 -0.55 0.21-0.45 -0.45 -0.45 -0.36 -0.36 -0.36 -0.48
-0.54 -0.18 -0.22 -0.12 0.110.24 0.24 0.24 0.05 0.05 0.05 0.07
0.17 0.03 0.12 0.09 0.00-0.07 -0.07 -0.07 -0.05 -0.05 -0.05 -0.06
0.00 3.52 5.05 -8.57 0.00 11.73 -1.45 -2.03 -8.24
AF BG0.00 0.001.90 -0.55-0.22 -0.180.12 0.03-0.03 -0.021.76 -0.73
6.89 -2.82
T U
2.90 2.90
O P4.22 -1.74
5.34 -2.23
H4
4.06 -1.44
O P
2.90 2.90
K L2.88 -1.11
4.29 -1.78
4.73 -1.93
K L
2.90 2.90
F G3.37 -1.37
5.05 -2.03
3.52 -1.45
F G
2.90 2.90
H1
H2
H3
A B1.82 -0.75
1.76 -0.73
ECUACIONES DE EQUILIBRIO
1ER NIVEL
R1
∑V + ∑H - R1 - R2
0.4 + 71.195 - R1 - R2
R1 + R2 + R3
2DO NIVEL
∑V + ∑H - R2 - R3
∑▒ 〖𝐹 _𝑋 = 0 〗 → ∑▒𝑉 + ∑▒𝐻 + ∑▒𝑅 = 0
∑▒ 〖𝐹 _ = 0𝑋 〗
∑▒ 〖𝐹 _𝑋 = 0 〗
0.67 + 47.463 - R2 - R3
R2 + R3 + R4
3ER NIVEL
∑V + ∑H - R3 - R4
0.57 + 26.105 - R3 - R4
R3 + R4 = 26.675
4TO NIVEL
∑V + ∑H - R4 = 0
0.97 + 9.493 - R4 = 0
R4 = 10.463
SOLUCION DE SISTEMA DE ECUACIONES
∑▒ 〖𝐹 _𝑋 = 0 〗
∑▒ 〖𝐹 _𝑋 = 0 〗
R4 = 10.463 Tn
R3 = 16.212 Tn
R2 = 21.458 Tn
R1 = 23.452 Tn
CROSS DESPLAZAMIENTO 1ER NIVEL
CON:
∴ M = -5
FACTORES DE DISTRIBUCIÓN
1ER NIVEL 1ER NIVEL 2DO NIVEL 3ER NIVEL
NUDO NUDO NUDO NUDO
A AF = 0.000F
FA = 0.333K
KF = 0.333O
OK = 0.333B BG = 0.000 FK = 0.333 KL = 0.335 OP = 0.335C CH = 0.000 FG = 0.334 KO = 0.332 OT = 0.332D DI = 0.000
G
GB = 0.230
L
LG = 0.230
P
PL = 0.230E EJ = 0.000 GF = 0.231 LK = 0.231 PO = 0.231
GL = 0.230 LP = 0.230 PU = 0.230GH = 0.309 LM = 0.309 PQ = 0.309
H
HC = 0.220
M
MH = 0.220
Q
QM = 0.220HG = 0.295 ML = 0.295 QP = 0.295HM = 0.220 MQ = 0.220 QV = 0.220HI = 0.265 MN = 0.265 QR = 0.265
I
ID = 0.220
N
NI = 0.220
R
RN = 0.220
FACTORES DE DISTRIBUCION
FACTORES DE DISTRIBUCION
FACTORES DE DISTRIBUCION
FACTORES DE DISTRIBUCION
- 6𝐸𝐼"Δ" /𝐿^2 =𝑀_((𝐴𝐿𝐸𝐴𝑇𝑂𝑅𝐼𝑂))
I IH = 0.265 N NM = 0.265 R RQ = 0.265IN = 0.220 NR = 0.220 RW = 0.220IJ = 0.295 NÑ = 0.295 RS = 0.295
JJE = 0.299
ÑÑJ = 0.299
SSÑ = 0.299
JI = 0.402 ÑN = 0.402 SR = 0.402JÑ = 0.299 ÑS = 0.299 SX = 0.299
TO TU UT UP UV0.498 0.502 0.301 0.299 0.400
0.00 0.00 0.00 0.00 0.000.00 0.00 0.00 0.00 0.000.00 0.00 0.00 0.00 0.000.14 0.00 0.00 0.00 0.00-0.07 -0.07 0.00 0.00 0.00-0.02 0.00 -0.03 -0.03 -0.010.01 0.01 0.02 0.02 0.03
0.00 0.06 -0.06 0.00 -0.01 0.00 0.02
OK OT OP PO PL PU PQ0.333 0.332 0.335 0.231 0.230 0.230 0.309
0.00 0.00 0.00 0.00 0.00 0.00 0.00-0.83 0.00 0.00 0.00 0.00 0.00 0.000.28 0.28 0.28 0.00 0.00 0.00 0.000.10 0.00 0.00 0.14 0.00 0.00 0.08-0.03 -0.03 -0.03 -0.05 -0.05 -0.05 -0.07-0.08 -0.03 -0.03 -0.02 -0.01 0.00 -0.030.05 0.05 0.05 0.01 0.01 0.01 0.02
0.00 -0.52 0.25 0.27 0.00 0.09 -0.05 -0.04 0.00
KO KF KL LK LP LG LM0.332 0.333 0.335 1 0.231 0.230 0.230 0.309 1
5.00 5.00-1.66 -1.67 -1.68 -1.16 -1.15 -1.15 -1.550.00 0.00 -0.58 -0.84 0.00 0.00 -0.740.19 0.19 0.19 0.36 0.36 0.36 0.490.14 0.14 0.18 0.10 0.00 0.07 0.21-0.15 -0.15 -0.15 -0.09 -0.09 -0.09 -0.12-0.02 -0.03 -0.04 -0.08 -0.03 -0.05 -0.09
0.03 0.03 0.03 0.05 0.05 0.05 0.070.00 -1.47 3.51 -2.05 0.00 -1.64 -0.85 4.20 -1.71
FA FK FG GF GB GL GH0.333 0.333 0.334 1 0.231 0.230 0.230 0.309 1-5.00 5.00 -5.00 5.000.00 0.00 0.00 0.00 0.00 0.00 0.00
-0.83 0.00 0.00 -0.58 0.000.28 0.28 0.28 0.13 0.13 0.13 0.18
0.10 0.07 0.14 0.18 0.08-0.05 -0.05 -0.05 -0.09 -0.09 -0.09 -0.12
-0.08 -0.05 -0.03 -0.04 -0.050.04 0.04 0.04 0.03 0.03 0.03 0.04
0.00 -4.74 4.45 0.28 0.00 0.18 -4.93 4.63 0.12
AF BG0.00 0.00-5.00 -5.000.00 0.000.14 0.07-0.03 -0.050.02 0.01-4.87 -4.97
0.06 0.00
T U
2.90 2.90
O P0.11 -0.01
0.25 -0.04
-0.52 -0.05
O P
2.90 2.90
K L-0.69 -0.31
-1.47 -0.85
3.51 4.20
K L
2.90 2.90
F G2.75 3.04
4.45 4.63
-4.74 -4.93
F G
2.90 2.90
A B-3.31 -3.41
-4.87 -4.97
ECUACIONES DE EQUILIBRIO
1ER NIVEL
∑V + Q11 - Q21
-16.9 + Q11 - Q21
- Q11 + Q21 + Q31
2DO NIVEL
∑V - Q21 - Q31
14.71 - Q21 - Q31
∑▒ 〖𝐹 _𝑋 = 0 〗 → ∑▒𝑉 + ∑▒𝑄 = 0
∑▒ 〖𝐹 _𝑋 = 0 〗
∑▒ 〖𝐹 _𝑋 = 0 〗
14.71 - Q21 - Q31
Q21 + Q31 + Q41
3ER NIVEL
∑V - Q31 - Q41
-2.42 - Q31 - Q41
Q31 + Q41 = -2.420
4TO NIVEL
∑V - Q41 = 0
0.25 - Q41 = 0
Q41 = 0.250
SOLUCION DE SISTEMA DE ECUACIONES
Q41 = 0.250 Tn
∑▒ 〖𝐹 _𝑋 = 0 〗
∑▒ 〖𝐹 _𝑋 = 0 〗
Q41 = 0.250 Tn
Q31 = -2.670 Tn
Q21 = 17.130 Tn
Q11 = 31.590 Tn
CROSS DESPLAZAMIENTO 2DO NIVEL
CON:
∴ M = -5
FACTORES DE DISTRIBUCIÓN
1ER NIVEL 1ER NIVEL 2DO NIVEL 3ER NIVEL
NUDO NUDO NUDO NUDO
A AF = 0.000F
FA = 0.333K
KF = 0.333O
OK = 0.333B BG = 0.000 FK = 0.333 KL = 0.335 OP = 0.335C CH = 0.000 FG = 0.334 KO = 0.332 OT = 0.332D DI = 0.000
G
GB = 0.230
L
LG = 0.230
P
PL = 0.230E EJ = 0.000 GF = 0.231 LK = 0.231 PO = 0.231
GL = 0.230 LP = 0.230 PU = 0.230GH = 0.309 LM = 0.309 PQ = 0.309
H
HC = 0.220
M
MH = 0.220
Q
QM = 0.220HG = 0.295 ML = 0.295 QP = 0.295HM = 0.220 MQ = 0.220 QV = 0.220HI = 0.265 MN = 0.265 QR = 0.265
I
ID = 0.220
N
NI = 0.220
R
RN = 0.220IH = 0.265 NM = 0.265 RQ = 0.265
FACTORES DE DISTRIBUCION
FACTORES DE DISTRIBUCION
FACTORES DE DISTRIBUCION
FACTORES DE DISTRIBUCION
- 6𝐸𝐼"Δ" /𝐿^2 =𝑀_((𝐴𝐿𝐸𝐴𝑇𝑂𝑅𝐼𝑂))
IIN = 0.220
NNR = 0.220
RRW = 0.220
IJ = 0.295 NÑ = 0.295 RS = 0.295
JJE = 0.299
ÑÑJ = 0.299
SSÑ = 0.299
JI = 0.402 ÑN = 0.402 SR = 0.402JÑ = 0.299 ÑS = 0.299 SX = 0.299
TO TU UT UP UV0.498 0.502 0.301 0.299 0.400
0.00 0.00 0.00 0.00 0.00-0.83 0.00 0.00 -0.58 0.000.41 0.42 0.17 0.17 0.230.10 0.09 0.21 0.21 0.10-0.09 -0.09 -0.16 -0.16 -0.21-0.07 -0.08 -0.05 -0.06 -0.070.07 0.07 0.05 0.05 0.07
0.00 -0.41 0.41 0.00 0.23 -0.35 0.12
OK OT OP PO PL PU PQ0.333 0.332 0.335 0.231 0.230 0.230 0.3095.00 5.00-1.67 -1.66 -1.68 -1.16 -1.15 -1.15 -1.550.00 0.00 -0.58 -0.84 -0.29 0.00 -0.740.19 0.19 0.19 0.43 0.43 0.43 0.580.00 0.21 0.22 0.10 0.11 0.09 0.21-0.14 -0.14 -0.14 -0.12 -0.12 -0.12 -0.150.00 -0.05 -0.06 -0.07 -0.03 -0.08 -0.090.03 0.03 0.03 0.06 0.06 0.06 0.08
0.00 3.42 -1.41 -2.01 0.00 -1.59 4.01 -0.77 -1.65
KO KF KL LK LP LG LM0.332 0.333 0.335 1 0.231 0.230 0.230 0.309 15.00 -5.00 5.00 -5.000.00 0.00 0.00 0.00 0.00 0.00 0.00-0.83 0.83 0.00 0.00 -0.58 0.58 0.000.00 0.00 0.00 0.00 0.00 0.00 0.000.10 -0.10 0.00 0.00 0.21 -0.18 0.000.00 0.00 0.00 -0.01 -0.01 -0.01 -0.01-0.07 0.03 0.00 0.00 -0.06 0.04 0.000.01 0.01 0.01 0.01 0.01 0.01 0.01
0.00 4.21 -4.22 0.01 0.00 0.00 4.58 -4.57 0.00
FA FK FG GF GB GL GH0.333 0.333 0.334 1 0.231 0.230 0.230 0.309 10.00 -5.00 0.00 -5.001.67 1.67 1.67 1.16 1.15 1.15 1.55
0.00 0.58 0.84 0.00 0.74-0.19 -0.19 -0.19 -0.36 -0.36 -0.36 -0.49
0.00 -0.18 -0.10 0.00 -0.210.06 0.06 0.06 0.07 0.07 0.07 0.10
0.00 0.04 0.03 0.00 0.07-0.01 -0.01 -0.01 -0.02 -0.02 -0.02 -0.03
0.00 1.52 -3.48 1.96 0.00 1.61 0.84 -4.17 1.72
AF BG0.00 0.000.00 0.000.83 0.58-0.10 -0.180.03 0.04-0.01 -0.010.76 0.42
-0.41 -0.35
T U
2.90 2.90
O P-0.63 -0.39
-1.41 -0.77
3.42 4.01
O P
2.90 2.90
K L2.63 2.96
4.21 4.58
-4.22 -4.57
K L
2.90 2.90
F G-2.65 -3.01
-3.48 -4.17
1.52 0.84
F G
2.90 2.90
A B0.79 0.43
0.76 0.42
ECUACIONES DE EQUILIBRIO
1ER NIVEL
∑V - Q12 + Q22
2.77 - Q12 + Q22
Q12 - Q22 + Q32
2DO NIVEL
∑V + Q22 - Q32
-14.5 + Q22 - Q32
∑▒ 〖𝐹 _𝑋 = 0 〗 → ∑▒𝑉 + ∑▒𝑄 = 0
∑▒ 〖𝐹 _𝑋 = 0 〗
∑▒ 〖𝐹 _𝑋 = 0 〗
- Q22 + Q32 + Q42
3ER NIVEL
∑V - Q32 - Q42
14.36 - Q32 - Q42
Q32 + Q42 = 14.360
4TO NIVEL
∑V - Q42 = 0
-2.38 - Q42 = 0
Q42 = -2.380
SOLUCION DE SISTEMA DE ECUACIONES
Q42 = -2.380 Tn
∑▒ 〖𝐹 _𝑋 = 0 〗
∑▒ 〖𝐹 _𝑋 = 0 〗
Q32 = 16.740 Tn
Q22 = 28.810 Tn
Q12 = 17.220 Tn
CROSS DESPLAZAMIENTO 3ER NIVEL
CON:
∴ M = -5
FACTORES DE DISTRIBUCIÓN
1ER NIVEL 1ER NIVEL 2DO NIVEL 3ER NIVEL
NUDO NUDO NUDO NUDO
A AF = 0.000F
FA = 0.333K
KF = 0.333O
OK = 0.333B BG = 0.000 FK = 0.333 KL = 0.335 OP = 0.335C CH = 0.000 FG = 0.334 KO = 0.332 OT = 0.332D DI = 0.000
G
GB = 0.230
L
LG = 0.230
P
PL = 0.230E EJ = 0.000 GF = 0.231 LK = 0.231 PO = 0.231
GL = 0.230 LP = 0.230 PU = 0.230GH = 0.309 LM = 0.309 PQ = 0.309
H
HC = 0.220
M
MH = 0.220
Q
QM = 0.220HG = 0.295 ML = 0.295 QP = 0.295HM = 0.220 MQ = 0.220 QV = 0.220HI = 0.265 MN = 0.265 QR = 0.265
I
ID = 0.220
N
NI = 0.220
R
RN = 0.220IH = 0.265 NM = 0.265 RQ = 0.265IN = 0.220 NR = 0.220 RW = 0.220
FACTORES DE DISTRIBUCION
FACTORES DE DISTRIBUCION
FACTORES DE DISTRIBUCION
FACTORES DE DISTRIBUCION
- 6𝐸𝐼"Δ" /𝐿^2 =𝑀_((𝐴𝐿𝐸𝐴𝑇𝑂𝑅𝐼𝑂))
I
IJ = 0.295
N
NÑ = 0.295
R
RS = 0.295
JJE = 0.299
ÑÑJ = 0.299
SSÑ = 0.299
JI = 0.402 ÑN = 0.402 SR = 0.402JÑ = 0.299 ÑS = 0.299 SX = 0.299
TO TU UT UP UV0.498 0.502 0.301 0.299 0.4005.00 5.00-2.49 -2.51 -1.51 -1.49 -2.000.00 -0.75 -1.26 0.00 -0.950.37 0.38 0.66 0.66 0.880.07 0.33 0.19 0.09 0.35-0.20 -0.20 -0.19 -0.19 -0.25-0.03 -0.09 -0.10 -0.05 -0.160.06 0.06 0.09 0.09 0.13
0.00 2.79 -2.79 0.00 -2.10 4.10 -2.00
OK OT OP PO PL PU PQ0.333 0.332 0.335 0.231 0.230 0.230 0.309-5.00 5.00 -5.00 5.000.00 0.00 0.00 0.00 0.00 0.00 0.000.83 -1.25 0.00 0.00 0.00 -0.75 0.000.14 0.14 0.14 0.17 0.17 0.17 0.23-0.10 0.19 0.09 0.07 0.04 0.33 0.02-0.06 -0.06 -0.06 -0.11 -0.11 -0.11 -0.140.04 -0.10 -0.05 -0.03 -0.03 -0.09 -0.040.04 0.04 0.04 0.04 0.04 0.04 0.06
0.00 -4.11 3.96 0.15 0.00 0.15 -4.88 4.60 0.13
KO KF KL LK LP LG LM0.332 0.333 0.335 1 0.231 0.230 0.230 0.309 1-5.00 0.00 -5.00 0.001.66 1.67 1.68 1.16 1.15 1.15 1.550.00 0.00 0.58 0.84 0.00 0.00 0.74-0.19 -0.19 -0.19 -0.36 -0.36 -0.36 -0.490.07 -0.14 -0.18 -0.10 0.09 -0.07 -0.210.08 0.08 0.08 0.07 0.07 0.07 0.09-0.03 0.03 0.03 0.04 -0.05 0.05 0.07-0.01 -0.01 -0.01 -0.03 -0.03 -0.03 -0.03
0.00 -3.42 1.43 1.98 0.00 1.62 -4.14 0.81 1.71
FA FK FG GF GB GL GH0.333 0.333 0.334 1 0.231 0.230 0.230 0.309 10.00 0.00 0.00 0.000.00 0.00 0.00 0.00 0.00 0.00 0.00
0.83 0.00 0.00 0.58 0.00-0.28 -0.28 -0.28 -0.13 -0.13 -0.13 -0.18
-0.10 -0.07 -0.14 -0.18 -0.080.05 0.05 0.05 0.09 0.09 0.09 0.12
0.04 0.05 0.03 0.03 0.05-0.03 -0.03 -0.03 -0.02 -0.02 -0.02 -0.03
0.00 -0.25 0.53 -0.27 0.00 -0.18 -0.06 0.36 -0.12
AF BG0.00 0.000.00 0.000.00 0.00-0.14 -0.070.03 0.05-0.01 -0.01-0.13 -0.03
2.79 4.10
T U
2.90 2.90
O P2.33 3
3.96 4.60
-4.11 -4.88
O P
2.90 2.90
K L-2.6 -3.11
-3.42 -4.14
1.43 0.81
K L
2.90 2.90
F G0.68 0.4
0.53 0.36
-0.25 -0.06
F G
2.90 2.90
A B-0.13 -0.03
-0.13 -0.03
ECUACIONES DE EQUILIBRIO
1ER NIVEL
∑V - Q13 - Q23
-0.34 - Q13 - Q23
Q13 + Q23 - Q33
2DO NIVEL
∑V - Q23 + Q33
2.49 - Q23 + Q33
∑▒ 〖𝐹 _𝑋 = 0 〗 → ∑▒𝑉 + ∑▒𝑄 = 0
∑▒ 〖𝐹 _𝑋 = 0 〗
∑▒ 〖𝐹 _𝑋 = 0 〗
+ Q23 - Q33 + Q43
3ER NIVEL
∑V + Q33 - Q43
-14.5 + Q33 - Q43
- Q33 + Q43 = -14.450
4TO NIVEL
∑V - Q43 = 0
13.72 - Q43 = 0
Q43 = 13.720
SOLUCION DE SISTEMA DE ECUACIONES
Q43 = 13.720 Tn
∑▒ 〖𝐹 _𝑋 = 0 〗
∑▒ 〖𝐹 _𝑋 = 0 〗
Q33 = 28.170 Tn
Q23 = 16.940 Tn
Q13 = -2.830 Tn
CROSS DESPLAZAMIENTO 4TO NIVEL
CON:
∴ M = -5
FACTORES DE DISTRIBUCIÓN
1ER NIVEL 1ER NIVEL 2DO NIVEL 3ER NIVEL
NUDO NUDO NUDO NUDO
A AF = 0.000F
FA = 0.333K
KF = 0.333O
OK = 0.333B BG = 0.000 FK = 0.333 KL = 0.335 OP = 0.335C CH = 0.000 FG = 0.334 KO = 0.332 OT = 0.332D DI = 0.000
G
GB = 0.230
L
LG = 0.230
P
PL = 0.230E EJ = 0.000 GF = 0.231 LK = 0.231 PO = 0.231
GL = 0.230 LP = 0.230 PU = 0.230GH = 0.309 LM = 0.309 PQ = 0.309
H
HC = 0.220
M
MH = 0.220
Q
QM = 0.220HG = 0.295 ML = 0.295 QP = 0.295HM = 0.220 MQ = 0.220 QV = 0.220HI = 0.265 MN = 0.265 QR = 0.265
I
ID = 0.220
N
NI = 0.220
R
RN = 0.220IH = 0.265 NM = 0.265 RQ = 0.265
FACTORES DE DISTRIBUCION
FACTORES DE DISTRIBUCION
FACTORES DE DISTRIBUCION
FACTORES DE DISTRIBUCION
- 6𝐸𝐼"Δ" /𝐿^2 =𝑀_((𝐴𝐿𝐸𝐴𝑇𝑂𝑅𝐼𝑂))
IIN = 0.220
NNR = 0.220
RRW = 0.220
IJ = 0.295 NÑ = 0.295 RS = 0.295
JJE = 0.299
ÑÑJ = 0.299
SSÑ = 0.299
JI = 0.402 ÑN = 0.402 SR = 0.402JÑ = 0.299 ÑS = 0.299 SX = 0.299
TO TU UT UP UV0.498 0.502 0.301 0.299 0.400-5.00 -5.002.49 2.51 1.51 1.49 2.000.83 0.75 1.26 0.58 0.95-0.79 -0.79 -0.84 -0.83 -1.11-0.30 -0.42 -0.40 -0.30 -0.450.36 0.36 0.35 0.34 0.460.14 0.17 0.18 0.14 0.24-0.16 -0.16 -0.17 -0.17 -0.22
0.00 -2.43 2.43 0.00 1.89 -3.75 1.86
OK OT OP PO PL PU PQ0.333 0.332 0.335 0.231 0.230 0.230 0.309
-5.00 -5.001.67 1.66 1.68 1.16 1.15 1.15 1.550.00 1.25 0.58 0.84 0.29 0.75 0.74-0.61 -0.61 -0.61 -0.60 -0.60 -0.60 -0.81-0.14 -0.39 -0.30 -0.31 -0.15 -0.41 -0.320.28 0.28 0.28 0.27 0.27 0.27 0.370.06 0.18 0.14 0.14 0.07 0.17 0.16-0.13 -0.13 -0.13 -0.13 -0.13 -0.13 -0.17
0.00 1.13 -2.76 1.63 0.00 1.37 0.90 -3.80 1.52
KO KF KL LK LP LG LM0.332 0.333 0.335 1 0.231 0.230 0.230 0.309 1
0.00 0.00 0.00 0.00 0.00 0.00 0.000.83 0.00 0.00 0.00 0.58 0.00 0.00-0.28 -0.28 -0.28 -0.13 -0.13 -0.13 -0.18-0.30 0.00 -0.07 -0.14 -0.30 0.00 -0.080.12 0.12 0.12 0.12 0.12 0.12 0.160.14 0.02 0.06 0.06 0.14 0.01 0.06-0.07 -0.07 -0.07 -0.06 -0.06 -0.06 -0.08
0.00 0.44 -0.20 -0.24 0.00 -0.15 0.34 -0.07 -0.12
FA FK FG GF GB GL GH0.333 0.333 0.334 1 0.231 0.230 0.230 0.309 1
0.00 0.00 0.00 0.00 0.00 0.00 0.000.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.00 0.00-0.14 0.00 0.00 -0.07 0.00
0.05 0.05 0.05 0.02 0.02 0.02 0.020.06 0.01 0.02 0.06 0.01
-0.02 -0.02 -0.02 -0.02 -0.02 -0.02 -0.030.00 0.02 -0.05 0.03 0.00 0.02 -0.01 -0.01 0.00
AF BG0.00 0.000.00 0.000.00 0.000.00 0.000.02 0.01-0.01 -0.010.01 0.00
-2.43 -3.75
T U
2.90 2.90
O P-1.79 -2.6
-2.76 -3.80
1.13 0.90
O P
2.90 2.90
K L0.54 0.43
0.44 0.34
-0.20 -0.07
K L
2.90 2.90
F G-0.09 -0.03
-0.05 -0.01
0.02 -0.01
F G
2.90 2.90
A B0.01 0
0.01 0.00
ECUACIONES DE EQUILIBRIO
1ER NIVEL
∑V - Q14 - Q24
0.02 - Q14 - Q24
Q14 + Q24 + Q34
2DO NIVEL
∑V - Q24 - Q34
-0.26 - Q24 - Q34
∑▒ 〖𝐹 _𝑋 = 0 〗 → ∑▒𝑉 + ∑▒𝑄 = 0
∑▒ 〖𝐹 _𝑋 = 0 〗
∑▒ 〖𝐹 _𝑋 = 0 〗
+ Q24 + Q34 - Q44
3ER NIVEL
∑V - Q34 + Q44
2.2 - Q34 + Q44
+ Q34 - Q44 = 2.200
4TO NIVEL
∑V + Q44 = 0
-11.6 + Q44 = 0
- Q43 = -11.560
SOLUCION DE SISTEMA DE ECUACIONES
∑▒ 〖𝐹 _𝑋 = 0 〗
∑▒ 〖𝐹 _𝑋 = 0 〗
Q44 = 11.560 Tn
Q34 = 13.760 Tn
Q24 = -2.460 Tn
Q14 = 0.280 Tn
CALCULO DE MOMENTOS FINALES
CUADRO DE MATRICES
i i1 23.452 31.590 17.220 -2.830 0.280 1 23.452 31.5902 21.458 17.130 28.810 16.940 -2.460 2 21.458 -17.1303 16.212 -2.670 16.740 28.170 13.760 3 16.212 2.6704 10.463 0.250 -2.380 13.720 11.560 4 10.463 -0.250
i1 2.0642 4.6293 6.643 MOMENTOS FINALES
Ri Qi1 Qi2 Qi3 Qi4 Ri Qi1
Ci
∑▒ 〖𝐹 _𝑋 = 0 〗 → −𝑅 _1+𝐶_1 𝑥𝑄_11−𝐶_2 𝑥𝑄_12− _𝐶 3 _1𝑥𝑄 3− _𝐶 4 _1𝑥𝑄 4 = 0∑▒ 〖𝐹 _𝑋 = 0 〗 → −𝑅 _2−𝐶_1 𝑥𝑄_21+𝐶_2 𝑥𝑄_22− _𝐶 3 _𝑥𝑄 23− _𝐶 4 _𝑥𝑄 24 = 0∑▒ 〖𝐹 _𝑋 = 0 〗 → −𝑅 _3−𝐶_1 𝑥𝑄_31−𝐶_2 𝑥𝑄_32+ _𝐶 3 _𝑥𝑄 33− _𝐶 4 _𝑥𝑄 34 = 0∑▒ 〖𝐹 _𝑋 = 0 〗 → −𝑅 _4−𝐶_1 𝑥𝑄_41−𝐶_2 𝑥𝑄_42− _𝐶 3 _𝑥𝑄 43+ _𝐶 4 _𝑥𝑄 44 = 0
4 7.881 MOMENTOS FINALES
4.50 -6.73
T U
2.90 2.90
O P
3.85 -5.27
0.46 -8.24
O P
2.90 2.90
K L
1.49 -7.16
0.37 -9.57
K LH2
H3
H4
2.90 2.90
F G
1.20 -9.45
-0.71 -8.24
F G
2.90 2.90
A B
-5.51 -9.06
H1
MÉTODO CROSS
2.90
2.90
2.90
2.90
TRAMO43.920 Tn/m3.920 Tn/m3.920 Tn/m3.920 Tn/m FACTORES DE DISTRIBUCIÓN
1ER NIVEL 1ER NIVEL 2DO NIVEL 3ER NIVEL
NUDO NUDO NUDO NUDO
A AF = 0.000F
FA = 0.333K
KF = 0.333OB BG = 0.000 FK = 0.333 KL = 0.335
C CH = 0.000 FG = 0.334 KO = 0.332D DI = 0.000
G
GB = 0.230
L
LG = 0.230
PE EJ = 0.000 GF = 0.231 LK = 0.231GL = 0.230 LP = 0.230GH = 0.309 LM = 0.309
H
HC = 0.220
M
MH = 0.220
QHG = 0.295 ML = 0.295HM = 0.220 MQ = 0.220HI = 0.265 MN = 0.265
I
ID = 0.220
N
NI = 0.220
RIH = 0.265 NM = 0.265IN = 0.220 NR = 0.220
= 4.312E-04 IJ = 0.295 NÑ = 0.295= 4.312E-04
JJE = 0.299
ÑÑJ = 0.299
S= 4.312E-04 JI = 0.402 ÑN = 0.402= 4.312E-04 JÑ = 0.299 ÑS = 0.299= 4.312E-04
= 4.340E-04= 5.787E-04= 5.208E-04= 5.787E-04
FACTORES DE DISTRIBUCION
FACTORES DE DISTRIBUCION
FACTORES DE DISTRIBUCION
FACTORES DE DISTRIBUCIÓN
3ER NIVEL 4TO NIVEL
NUDO
0.333 T TO = 0.4980.335 TU = 0.5020.332
UUP = 0.299
0.230 UT = 0.3010.231 UV = 0.4000.230
VVQ = 0.282
0.309 VU = 0.3780.220 VW = 0.3400.295
WWR = 0.282
0.220 WV = 0.340
FACTORES DE DISTRIBUCION
FACTORES DE DISTRIBUCION
0.265W
WX = 0.3780.220 X XS = 0.4270.265 XW = 0.5730.2200.2950.2990.4020.299
VU VQ VW WV WR WX0.378 0.282 0.340 0.340 0.282 0.3786.61 -8.07 8.07 -6.610.55 0.41 0.49 -0.49 -0.41 -0.55-0.96 0.16 -0.25 0.25 -0.16 -1.900.39 0.29 0.35 0.61 0.51 0.68-0.52 0.06 0.31 0.18 0.17 0.360.06 0.04 0.05 -0.24 -0.20 -0.270.05 -0.03 -0.12 0.03 -0.11 -0.210.04 0.03 0.03 0.10 0.08 0.11
0.00 6.23 0.97 -7.20 0.00 8.50 -0.12 -8.38
QP QM QV QR RQ RN RW RS0.295 0.220 0.220 0.265 0.265 0.220 0.220 0.2956.61 -8.07 8.07 -6.610.43 0.32 0.32 0.39 -0.39 -0.32 -0.32 -0.43-0.74 0.16 0.20 -0.19 0.19 -0.16 -0.20 -1.330.17 0.12 0.12 0.15 0.40 0.33 0.33 0.44-0.18 0.07 0.15 0.20 0.07 0.16 0.25 0.53-0.07 -0.05 -0.05 -0.06 -0.27 -0.22 -0.22 -0.300.16 -0.02 0.02 -0.13 -0.03 -0.09 -0.10 -0.16-0.01 -0.01 -0.01 -0.01 0.10 0.09 0.09 0.11
0.00 6.38 0.59 0.76 -7.73 0.00 8.15 -0.22 -0.18 -7.75 0.00
ML MQ MH MN NM NR NI NÑ0.295 0.220 0.220 0.265 1 0.265 0.220 0.220 0.295 16.61 -8.07 8.07 -6.610.43 0.32 0.32 0.39 -0.39 -0.32 -0.32 -0.43-0.74 0.16 0.16 -0.19 0.19 -0.16 -0.16 -1.330.18 0.13 0.13 0.16 0.39 0.32 0.32 0.43-0.16 0.06 0.08 0.19 0.08 0.17 0.14 0.44
-0.05 -0.04 -0.04 -0.05 -0.22 -0.18 -0.18 -0.240.12 -0.03 0.00 -0.11 -0.02 -0.11 -0.06 -0.160.01 0.00 0.00 0.00 0.09 0.08 0.08 0.10
0.00 6.40 0.61 0.66 -7.67 0.00 8.19 -0.21 -0.18 -7.80 0.00
HG HC HM HI IH ID IN IJ0.295 0.220 0.220 0.265 1 0.265 0.220 0.220 0.295 16.61 -8.07 8.07 -6.610.43 0.32 0.32 0.39 -0.39 -0.32 -0.32 -0.43-0.74 0.16 -0.19 0.19 -0.16 -1.330.23 0.17 0.17 0.20 0.34 0.29 0.29 0.38-0.24 0.07 0.17 0.10 0.16 0.240.00 0.00 0.00 0.00 -0.13 -0.11 -0.11 -0.150.04 -0.02 -0.07 0.00 -0.09 -0.100.02 0.01 0.01 0.01 0.05 0.04 0.04 0.06
0.00 6.34 0.50 0.71 -7.55 0.00 8.24 -0.10 -0.19 -7.94 0.00
CH DI0.00 0.000.16 -0.160.08 0.140.00 -0.060.01 0.020.25 -0.05
0.97 -0.12
V W
2.90 2.90
Q R0.6 -0.1
0.76 -0.18
0.59 -0.22
Q R
2.90 2.90
M N0.42 -0.15
0.61 -0.21
0.66 -0.18
M N
2.90 2.90
H I0.47 -0.13
0.71 -0.19
0.50 -0.10
H I
2.90 2.90
C D0.26 -0.05
0.25 -0.05
- R3 - R4 = 0
- R3 - R4 = 0
+ R4 = 71.585
- R4 = 0
∑▒ 〖𝐹 _𝑋 = 0 〗 → ∑▒𝑉 + ∑▒𝐻 + ∑▒𝑅 = 0
- R4 = 0
= 48.133
= 0
= 0
26.675
FACTORES DE DISTRIBUCIÓN
3ER NIVEL 4TO NIVEL
NUDO
0.333 T TO = 0.4980.335 TU = 0.5020.332
UUP = 0.299
0.230 UT = 0.3010.231 UV = 0.4000.230
VVQ = 0.282
0.309 VU = 0.3780.220 VW = 0.3400.295
WWR = 0.282
0.220 WV = 0.3400.265 WX = 0.3780.220 X XS = 0.427
FACTORES DE DISTRIBUCION
FACTORES DE DISTRIBUCION
0.265X
XW = 0.5730.2200.2950.2990.4020.299
VU VQ VW WV WR WX0.378 0.282 0.340 0.340 0.282 0.378
0.00 0.00 0.00 0.00 0.00 0.000.00 0.00 0.00 0.00 0.00 0.000.00 0.00 0.00 0.00 0.00 0.000.00 0.06 0.00 0.00 0.06 0.00-0.02 -0.02 -0.02 -0.02 -0.02 -0.020.00 -0.03 -0.01 -0.01 -0.04 -0.030.01 0.01 0.01 0.03 0.02 0.03
0.00 -0.01 0.03 -0.02 0.00 0.00 0.02 -0.02
QP QM QV QR RQ RN RW RS0.295 0.220 0.220 0.265 0.265 0.220 0.220 0.295
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000.00 -0.55 0.00 0.00 0.00 -0.55 0.00 0.000.16 0.12 0.12 0.15 0.15 0.12 0.12 0.160.00 0.16 0.00 0.07 0.07 0.18 0.00 0.15-0.07 -0.05 -0.05 -0.06 -0.11 -0.09 -0.09 -0.12-0.03 -0.06 -0.01 -0.05 -0.03 -0.05 -0.01 -0.040.05 0.04 0.04 0.04 0.03 0.03 0.03 0.04
0.00 0.11 -0.35 0.10 0.15 0.00 0.11 -0.36 0.05 0.19 0.00
ML MQ MH MN NM NR NI NÑ0.295 0.220 0.220 0.265 1 0.265 0.220 0.220 0.295 1
5.00 5.00-1.47 -1.10 -1.10 -1.33 -1.33 -1.10 -1.10 -1.48-0.77 0.00 0.00 -0.66 -0.66 0.00 0.00 -1.010.42 0.32 0.32 0.38 0.44 0.37 0.37 0.490.24 0.06 0.06 0.22 0.19 0.06 0.06 0.15-0.17 -0.13 -0.13 -0.16 -0.12 -0.10 -0.10 -0.14-0.06 -0.03 -0.04 -0.06 -0.08 -0.04 -0.04 -0.09
0.05 0.04 0.04 0.05 0.07 0.06 0.06 0.080.00 -1.76 -0.84 4.15 -1.55 0.00 -1.49 -0.76 4.24 -1.99 0.00
HG HC HM HI IH ID IN IJ0.295 0.220 0.220 0.265 1 0.265 0.220 0.220 0.295 1
-5.00 5.00 -5.00 5.000.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000.00 -0.55 0.00 0.00 -0.55 0.000.16 0.12 0.12 0.15 0.15 0.12 0.12 0.160.09 0.16 0.07 0.07 0.18 0.15-0.09 -0.07 -0.07 -0.08 -0.11 -0.09 -0.09 -0.12-0.06 -0.06 -0.05 -0.04 -0.05 -0.040.05 0.04 0.04 0.05 0.03 0.03 0.03 0.04
0.00 0.15 -4.91 4.63 0.13 0.00 0.10 -4.94 4.64 0.19 0.00
CH DI0.00 0.00-5.00 -5.000.00 0.000.06 0.06-0.04 -0.040.02 0.01-4.95 -4.97
0.03 0.02
V W
2.90 2.90
Q R0.04 0.03
0.10 0.05
-0.35 -0.36
Q R
2.90 2.90
M N-0.41 -0.39
-0.84 -0.76
4.15 4.24
M N
2.90 2.90
H I3.03 3.06
4.63 4.64
-4.91 -4.94
H I
2.90 2.90
C D-3.4 -3.42
-4.95 -4.97
- Q31 - Q41 = 0
- Q31 - Q41 = 0
+ Q41 = -16.880
- Q41 = 0
- Q41 = 0
∑▒ 〖𝐹 _𝑋 = 0 〗 → ∑▒𝑉 + ∑▒𝑄 = 0
- Q41 = 0
= 14.710
= 0
= 0
-2.420
FACTORES DE DISTRIBUCIÓN
3ER NIVEL 4TO NIVEL
NUDO
0.333 T TO = 0.4980.335 TU = 0.5020.332
UUP = 0.299
0.230 UT = 0.3010.231 UV = 0.4000.230
VVQ = 0.282
0.309 VU = 0.3780.220 VW = 0.3400.295
WWR = 0.282
0.220 WV = 0.3400.265 WX = 0.3780.220 X XS = 0.4270.265 XW = 0.573
FACTORES DE DISTRIBUCION
FACTORES DE DISTRIBUCION
0.2200.2950.2990.4020.299
VU VQ VW WV WR WX0.378 0.282 0.340 0.340 0.282 0.378
0.00 0.00 0.00 0.00 0.00 0.000.00 -0.55 0.00 0.00 -0.55 0.000.21 0.16 0.19 0.19 0.16 0.210.12 0.16 0.09 0.09 0.18 0.21-0.14 -0.10 -0.12 -0.17 -0.14 -0.19-0.11 -0.06 -0.08 -0.06 -0.05 -0.060.10 0.07 0.09 0.06 0.05 0.06
0.00 0.17 -0.33 0.16 0.00 0.11 -0.35 0.24
QP QM QV QR RQ RN RW RS0.295 0.220 0.220 0.265 0.265 0.220 0.220 0.295
5.00 5.00-1.47 -1.10 -1.10 -1.33 -1.33 -1.10 -1.10 -1.48-0.77 0.00 0.00 -0.66 -0.66 0.00 0.00 -1.010.42 0.32 0.32 0.38 0.44 0.37 0.37 0.490.29 0.00 0.08 0.22 0.19 0.00 0.08 0.15-0.17 -0.13 -0.13 -0.16 -0.11 -0.09 -0.09 -0.12-0.08 0.00 -0.05 -0.06 -0.08 0.00 -0.07 -0.080.05 0.04 0.04 0.05 0.06 0.05 0.05 0.07
0.00 -1.73 4.13 -0.85 -1.55 0.00 -1.48 4.23 -0.77 -1.98 0.00
ML MQ MH MN NM NR NI NÑ0.295 0.220 0.220 0.265 1 0.265 0.220 0.220 0.295 1
5.00 -5.00 5.00 -5.000.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000.00 -0.55 0.55 0.00 0.00 -0.55 0.55 0.000.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000.00 0.16 -0.16 0.00 0.00 0.18 -0.18 0.000.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00-0.01 -0.06 0.05 0.00 0.00 -0.05 0.04 0.000.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.00 0.00 4.55 -4.55 0.00 0.00 0.00 4.59 -4.59 0.00 0.00
HG HC HM HI IH ID IN IJ0.295 0.220 0.220 0.265 1 0.265 0.220 0.220 0.295 1
0.00 -5.00 0.00 -5.001.47 1.10 1.10 1.33 1.33 1.10 1.10 1.480.77 0.00 0.66 0.66 0.00 1.01-0.42 -0.32 -0.32 -0.38 -0.44 -0.37 -0.37 -0.49-0.24 0.00 -0.22 -0.19 0.00 -0.150.14 0.10 0.10 0.12 0.09 0.07 0.07 0.100.05 0.00 0.04 0.06 0.00 0.05-0.03 -0.02 -0.02 -0.02 -0.03 -0.02 -0.02 -0.03
0.00 1.74 0.87 -4.13 1.53 0.00 1.48 0.78 -4.22 1.96 0.00
CH DI0.00 0.000.00 0.000.55 0.55-0.16 -0.180.05 0.04-0.01 -0.010.43 0.39
-0.33 -0.35
V W
2.90 2.90
Q R-0.41 -0.38
-0.85 -0.77
4.13 4.23
Q R
2.90 2.90
M N2.99 3.04
4.55 4.59
-4.55 -4.59
M N
2.90 2.90
H I-3 -3.04
-4.13 -4.22
0.87 0.78
H I
2.90 2.90
C D0.45 0.41
0.43 0.39
- Q32 - Q42 = 0
- Q32 - Q42 = 0
+ Q42 = 2.770
- Q42 = 0
- Q42 = 0
∑▒ 〖𝐹 _𝑋 = 0 〗 → ∑▒𝑉 + ∑▒𝑄 = 0
= -14.450
= 0
= 0
14.360
FACTORES DE DISTRIBUCIÓN
3ER NIVEL 4TO NIVEL
NUDO
0.333 T TO = 0.4980.335 TU = 0.5020.332
UUP = 0.299
0.230 UT = 0.3010.231 UV = 0.4000.230
VVQ = 0.282
0.309 VU = 0.3780.220 VW = 0.3400.295
WWR = 0.282
0.220 WV = 0.3400.265 WX = 0.3780.220 X XS = 0.4270.265 XW = 0.5730.220
FACTORES DE DISTRIBUCION
FACTORES DE DISTRIBUCION
0.2950.2990.4020.299
VU VQ VW WV WR WX0.378 0.282 0.340 0.340 0.282 0.378
5.00 5.00-1.89 -1.41 -1.70 -1.70 -1.41 -1.89-1.00 0.00 -0.85 -0.85 0.00 -1.430.70 0.52 0.63 0.78 0.64 0.860.44 0.02 0.39 0.31 0.02 0.27-0.32 -0.24 -0.29 -0.20 -0.17 -0.23-0.12 -0.03 -0.10 -0.14 -0.02 -0.140.10 0.07 0.09 0.10 0.09 0.12
0.00 -2.10 3.94 -1.84 0.00 -1.70 4.14 -2.44
QP QM QV QR RQ RN RW RS0.295 0.220 0.220 0.265 0.265 0.220 0.220 0.295
-5.00 5.00 -5.00 5.000.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000.00 0.55 -0.71 0.00 0.00 0.55 -0.71 0.000.05 0.03 0.03 0.04 0.04 0.03 0.03 0.050.12 -0.16 0.26 0.02 0.02 -0.18 0.32 0.06-0.07 -0.05 -0.05 -0.06 -0.06 -0.05 -0.05 -0.07-0.07 0.06 -0.12 -0.03 -0.03 0.04 -0.08 -0.020.05 0.04 0.04 0.04 0.03 0.02 0.02 0.03
0.00 0.07 -4.53 4.45 0.01 0.00 0.00 -4.58 4.54 0.05 0.00
ML MQ MH MN NM NR NI NÑ0.295 0.220 0.220 0.265 1 0.265 0.220 0.220 0.295 1
-5.00 0.00 -5.00 0.001.47 1.10 1.10 1.33 1.33 1.10 1.10 1.480.77 0.00 0.00 0.66 0.66 0.00 0.00 1.01-0.42 -0.32 -0.32 -0.38 -0.44 -0.37 -0.37 -0.49-0.24 0.02 -0.06 -0.22 -0.19 0.02 -0.06 -0.150.15 0.11 0.11 0.13 0.10 0.08 0.08 0.110.04 -0.03 0.04 0.05 0.07 -0.02 0.04 0.06-0.03 -0.02 -0.02 -0.03 -0.04 -0.03 -0.03 -0.04
0.00 1.74 -4.14 0.85 1.54 0.00 1.48 -4.22 0.77 1.97 0.00
HG HC HM HI IH ID IN IJ0.295 0.220 0.220 0.265 1 0.265 0.220 0.220 0.295 1
0.00 0.00 0.00 0.000.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000.00 0.55 0.00 0.00 0.55 0.00-0.16 -0.12 -0.12 -0.15 -0.15 -0.12 -0.12 -0.16-0.09 -0.16 -0.07 -0.07 -0.18 -0.150.09 0.07 0.07 0.08 0.11 0.09 0.09 0.120.06 0.06 0.05 0.04 0.04 0.04-0.05 -0.04 -0.04 -0.05 -0.03 -0.03 -0.03 -0.04
0.00 -0.15 -0.09 0.36 -0.13 0.00 -0.10 -0.06 0.35 -0.19 0.00
CH DI0.00 0.000.00 0.000.00 0.00-0.06 -0.060.04 0.04-0.02 -0.01-0.04 -0.03
3.94 4.14
V W
2.90 2.90
Q R2.89 2.99
4.45 4.54
-4.53 -4.58
Q R
2.90 2.90
M N-2.99 -3.04
-4.14 -4.22
0.85 0.77
M N
2.90 2.90
H I0.42 0.39
0.36 0.35
-0.09 -0.06
H I
2.90 2.90
C D-0.05 -0.03
-0.04 -0.03
+ Q33 - Q43 = 0
+ Q33 - Q43 = 0
+ Q43 = -0.340
- Q43 = 0
- Q43 = 0
∑▒ 〖𝐹 _𝑋 = 0 〗 → ∑▒𝑉 + ∑▒𝑄 = 0
= 2.490
= 0
= 0
-14.450
FACTORES DE DISTRIBUCIÓN
3ER NIVEL 4TO NIVEL
NUDO
0.333 T TO = 0.4980.335 TU = 0.5020.332
UUP = 0.299
0.230 UT = 0.3010.231 UV = 0.4000.230
VVQ = 0.282
0.309 VU = 0.3780.220 VW = 0.3400.295
WWR = 0.282
0.220 WV = 0.3400.265 WX = 0.3780.220 X XS = 0.4270.265 XW = 0.573
FACTORES DE DISTRIBUCION
FACTORES DE DISTRIBUCION
0.2200.2950.2990.4020.299
VU VQ VW WV WR WX0.378 0.282 0.340 0.340 0.282 0.378
-5.00 -5.001.89 1.41 1.70 1.70 1.41 1.891.00 0.55 0.85 0.85 0.55 1.43-0.91 -0.68 -0.82 -0.96 -0.80 -1.07-0.56 -0.24 -0.48 -0.41 -0.26 -0.480.48 0.36 0.43 0.39 0.33 0.440.23 0.12 0.20 0.22 0.12 0.23-0.21 -0.15 -0.19 -0.19 -0.16 -0.21
0.00 1.93 -3.63 1.69 0.00 1.59 -3.81 2.22
QP QM QV QR RQ RN RW RS0.295 0.220 0.220 0.265 0.265 0.220 0.220 0.295
-5.00 -5.001.47 1.10 1.10 1.33 1.33 1.10 1.10 1.480.77 0.00 0.71 0.66 0.66 0.00 0.71 1.01-0.63 -0.47 -0.47 -0.57 -0.63 -0.52 -0.52 -0.70-0.40 -0.06 -0.34 -0.31 -0.28 -0.06 -0.40 -0.360.33 0.25 0.25 0.30 0.29 0.24 0.24 0.330.18 0.04 0.18 0.15 0.15 0.05 0.16 0.17-0.16 -0.12 -0.12 -0.15 -0.14 -0.12 -0.12 -0.16
0.00 1.56 0.74 -3.70 1.40 0.00 1.38 0.70 -3.83 1.75 0.00
ML MQ MH MN NM NR NI NÑ0.295 0.220 0.220 0.265 1 0.265 0.220 0.220 0.295 1
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000.00 0.55 0.00 0.00 0.00 0.55 0.00 0.00-0.16 -0.12 -0.12 -0.15 -0.15 -0.12 -0.12 -0.16-0.09 -0.24 0.00 -0.07 -0.07 -0.26 0.00 -0.150.12 0.09 0.09 0.11 0.13 0.11 0.11 0.140.08 0.12 0.01 0.06 0.05 0.12 0.01 0.07-0.08 -0.06 -0.06 -0.07 -0.07 -0.06 -0.06 -0.07
0.00 -0.13 0.34 -0.09 -0.12 0.00 -0.10 0.34 -0.06 -0.17 0.00
HG HC HM HI IH ID IN IJ0.295 0.220 0.220 0.265 1 0.265 0.220 0.220 0.295 1
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000.00 0.00 0.00 0.00 0.00 0.000.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000.00 -0.06 0.00 0.00 -0.06 0.000.02 0.01 0.01 0.02 0.02 0.01 0.01 0.020.01 0.04 0.01 0.01 0.05 0.02-0.02 -0.01 -0.01 -0.02 -0.02 -0.02 -0.02 -0.02
0.00 0.01 0.00 -0.02 0.01 0.00 0.00 -0.01 -0.01 0.02 0.00
CH DI0.00 0.000.00 0.000.00 0.000.00 0.000.01 0.01-0.01 -0.010.00 0.00
-3.63 -3.81
V W
2.90 2.90
Q R-2.53 -2.63
-3.70 -3.83
0.74 0.70
Q R
2.90 2.90
M N0.37 0.36
0.34 0.34
-0.09 -0.06
M N
2.90 2.90
H I-0.04 -0.03
-0.02 -0.01
0.00 -0.01
H I
2.90 2.90
C D0 0
0.00 0.00
- Q34 + Q44 = 0
- Q34 + Q44 = 0
- Q44 = 0.020
+ Q44 = 0
+ Q44 = 0
∑▒ 〖𝐹 _𝑋 = 0 〗 → ∑▒𝑉 + ∑▒𝑄 = 0
= -0.260
= 0
= 0
2.200
CALCULO DE MOMENTOS FINALES
CUADRO DE MATRICES MATRIZ INVERSA
i31.590 -17.220 2.830 -0.280 1 0.06220068 0.06640481 0.06639931 0.06638208-17.130 28.810 -16.940 2.460 2 0.06665521 0.13772130 0.14221141 0.142235522.670 -16.740 28.170 -13.760 3 0.06669326 0.14208038 0.21334627 0.21794751-0.250 2.380 -13.720 11.560 4 0.06663640 0.14204137 0.21774988 0.28945699
MOMENTOS FINALES
Qi1 Qi2 Qi3 Qi4 Qi1 Qi2 Qi3 Qi4
∑▒ 〖𝐹 _𝑋 = 0 〗 → −𝑅 _1+𝐶_1 𝑥𝑄_11−𝐶_2 𝑥𝑄_12− _𝐶 3 _1𝑥𝑄 3− _𝐶 4 _1𝑥𝑄 4 = 0∑▒ 〖𝐹 _𝑋 = 0 〗 → −𝑅 _2−𝐶_1 𝑥𝑄_21+𝐶_2 𝑥𝑄_22− _𝐶 3 _𝑥𝑄 23− _𝐶 4 _𝑥𝑄 24 = 0∑▒ 〖𝐹 _𝑋 = 0 〗 → −𝑅 _3−𝐶_1 𝑥𝑄_31−𝐶_2 𝑥𝑄_32+ _𝐶 3 _𝑥𝑄 33− _𝐶 4 _𝑥𝑄 34 = 0∑▒ 〖𝐹 _𝑋 = 0 〗 → −𝑅 _4−𝐶_1 𝑥𝑄_41−𝐶_2 𝑥𝑄_42− _𝐶 3 _𝑥𝑄 43+ _𝐶 4 _𝑥𝑄 44 = 0
MOMENTOS FINALES
-2.94 -4.22
V W
2.90 2.90
Q R
-2.54 -3.63
-5.35 -6.36
Q R
2.90 2.90
M N
-4.84 -5.91
-6.90 -8.09
M N
2.90 2.90
H I
-6.61 -7.90
-6.21 -7.10
H I
2.90 2.90
C D
-7.97 -8.52
FACTORES DE DISTRIBUCIÓN
3ER NIVEL 4TO NIVEL
NUDO
OK = 0.333 T TO = 0.498OP = 0.335 TU = 0.502OT = 0.332
UUP = 0.299
PL = 0.230 UT = 0.301PO = 0.231 UV = 0.400PU = 0.230
VVQ = 0.282
PQ = 0.309 VU = 0.378QM = 0.220 VW = 0.340QP = 0.295
WWR = 0.282
QV = 0.220 WV = 0.340QR = 0.265 WX = 0.378RN = 0.220 X XS = 0.427RQ = 0.265 XW = 0.573RW = 0.220RS = 0.295SÑ = 0.299SR = 0.402SX = 0.299
FACTORES DE DISTRIBUCION
FACTORES DE DISTRIBUCION
XW XS0.573 0.4276.61-3.79 -2.82-0.27 -0.990.72 0.540.34 0.39-0.42 -0.31-0.13 -0.120.15 0.11
0.00 3.21 -3.21
SR SÑ SX0.402 0.299 0.299 16.61-2.66 -1.98 -1.98-0.21 -0.99 -1.411.05 0.78 0.780.22 0.33 0.27-0.33 -0.24 -0.24-0.15 -0.12 -0.160.17 0.13 0.134.71 -2.09 -2.61
ÑN ÑS ÑJ0.402 0.299 0.299 16.61-2.66 -1.98 -1.98-0.21 -0.99 -0.990.88 0.66 0.660.21 0.39 0.18
-0.32 -0.23 -0.23-0.12 -0.12 -0.080.13 0.10 0.104.53 -2.18 -2.35
JI JE JÑ0.402 0.299 0.299 16.61-2.66 -1.98 -1.98-0.21 -0.990.48 0.36 0.360.19 0.33-0.21 -0.16 -0.16-0.07 -0.120.08 0.06 0.064.21 -1.72 -2.49
EJ0.00-0.990.18-0.080.03-0.86
-3.21
- X
2.90
S-2.01
-2.61
R4
-2.09
S
2.90
Ñ-1.47
-2.18
-2.35
Ñ
2.90
J-1.67
-2.49
-1.72
J
2.90
R1
R 2
R3
+E-0.89
-0.86
XW XS0.573 0.427
0.00 0.000.00 0.000.00 0.000.00 0.11-0.06 -0.05-0.01 -0.030.02 0.02
0.00 -0.05 0.05
SR SÑ SX0.402 0.299 0.299 1
0.00 0.00 0.000.00 -0.75 0.000.30 0.22 0.220.08 0.11 0.00-0.08 -0.06 -0.06-0.06 -0.07 -0.020.06 0.05 0.050.31 -0.50 0.19
ÑN ÑS ÑJ0.402 0.299 0.299 1
5.00-2.01 -1.50 -1.49-0.74 0.00 0.000.30 0.22 0.220.25 0.11 0.11-0.19 -0.14 -0.14-0.07 -0.03 -0.03
0.05 0.04 0.04-2.41 -1.29 3.71
JI JE JÑ0.402 0.299 0.299 1
-5.00 5.000.00 0.00 0.000.00 -0.750.30 0.22 0.220.08 0.11-0.08 -0.06 -0.06-0.06 -0.070.05 0.04 0.040.30 -4.79 4.50
EJ0.00-5.000.000.11-0.030.02-4.90
0.05
- X
2.90
S0.08
0.19
Q41
-0.50
S
2.90
Ñ-0.62
-1.29
3.71
Ñ
2.90
J2.83
4.50
-4.79
J
2.90
Q31
Q21
Q11
E-3.34
-4.90
XW XS0.573 0.427
0.00 0.000.00 -0.750.43 0.320.10 0.11-0.12 -0.09-0.09 -0.060.09 0.07
0.00 0.40 -0.40
SR SÑ SX0.402 0.299 0.299 1
5.00-2.01 -1.49 -1.50-0.74 0.00 0.000.30 0.22 0.220.25 0.00 0.16-0.16 -0.12 -0.12-0.06 0.00 -0.050.04 0.03 0.03-2.39 3.64 -1.25
ÑN ÑS ÑJ0.402 0.299 0.299 1
5.00 -5.000.00 0.00 0.000.00 -0.75 0.750.00 0.00 0.000.00 0.11 -0.110.00 0.00 0.000.00 -0.06 0.040.01 0.01 0.01
0.01 4.31 -4.32
JI JE JÑ0.402 0.299 0.299 1
0.00 -5.002.01 1.49 1.500.74 0.00-0.30 -0.22 -0.22-0.25 0.000.10 0.07 0.070.05 0.00-0.02 -0.01 -0.012.33 1.33 -3.67
EJ0.000.000.75-0.110.04-0.010.67
-0.40
- X
2.90
S-0.57
-1.25
Q42
3.64
S
2.90
Ñ2.74
4.31
-4.32
Ñ
2.90
J-2.75
-3.67
1.33
J
2.90
Q32
Q12
Q22
E0.69
0.67
XW XS0.573 0.427
5.00-2.86 -2.14-0.94 0.000.54 0.400.43 0.05-0.27 -0.20-0.11 -0.020.08 0.06
0.00 -3.15 3.15
SR SÑ SX0.402 0.299 0.299
-5.00 5.000.00 0.00 0.000.00 0.75 -1.070.13 0.10 0.100.02 -0.11 0.20-0.05 -0.03 -0.03-0.03 0.05 -0.100.04 0.03 0.030.11 -4.23 4.12
ÑN ÑS ÑJ0.402 0.299 0.299 1
-5.00 0.002.01 1.50 1.490.74 0.00 0.00-0.30 -0.22 -0.22-0.25 0.05 -0.110.12 0.09 0.090.06 -0.02 0.03-0.03 -0.02 -0.022.36 -3.62 1.26
JI JE JÑ0.402 0.299 0.299 1
0.00 0.000.00 0.00 0.000.00 0.75-0.30 -0.22 -0.22-0.08 -0.110.08 0.06 0.060.06 0.05-0.04 -0.03 -0.03-0.29 -0.20 0.49
EJ0.000.000.00-0.110.03-0.02-0.10
3.15
- X
2.90
S2.51
4.12
-4.23
Q43
S
2.90
Ñ-2.71
-3.62
1.26
Ñ
2.90
J0.6
0.49
-0.20
J
2.90
Q23
Q13
Q33
E-0.1
-0.10
XW XS0.573 0.427
-5.002.86 2.140.94 0.75-0.97 -0.72-0.54 -0.270.46 0.340.22 0.12-0.20 -0.15
0.00 2.79 -2.79
SR SÑ SX0.402 0.299 0.299
-5.002.01 1.49 1.500.74 0.00 1.07-0.73 -0.54 -0.54-0.35 -0.11 -0.360.33 0.25 0.250.16 0.05 0.17-0.16 -0.12 -0.122.01 1.03 -3.04
ÑN ÑS ÑJ0.402 0.299 0.299 1
0.00 0.00 0.000.00 0.75 0.00-0.30 -0.22 -0.22-0.08 -0.27 0.000.14 0.10 0.100.07 0.12 0.02-0.08 -0.06 -0.06
-0.25 0.42 -0.16
JI JE JÑ0.402 0.299 0.299 1
0.00 0.00 0.000.00 0.000.00 0.00 0.000.00 -0.110.04 0.03 0.030.01 0.05-0.02 -0.02 -0.020.03 0.02 -0.04
EJ0.000.000.000.000.02-0.010.01
-2.79
- X
2.90
S-2.01
-3.04
Q44
1.03
S
2.90
Ñ0.5
0.42
-0.16
Ñ
2.90
J-0.07
-0.04
0.02
J
2.90
Q24
Q34
Q14
E0.01
0.01
MATRIZ INVERSA 0 0 0 0.1 2.0640 0 0 0.1 4.6290 0 0 0.2 6.643
0.06638208 0 0 0 0.3 7.8810.142235520.217947510.28945699
Qi4
-6.02MOMENTOS FINALES
- X MAF = -5.51 MKF = 0.37MBG = -9.06 MKO = 1.49MCH = -7.97 MKL = -1.86MDI = -8.52 MLK = 17.84MEJ = -7.82 MLP = -7.16
2.90 MFA = -0.71 MLG = -9.57MFK = 1.20 0 MLM = -1.10MFG = -0.49 MML = 13.30MGF = 18.51 MMQ = -4.84MGL = -9.45 0 MMH = -6.90
S MGB = -8.24 MMN -1.57MGH = -0.82 MNM = 14.17MHG = 13.81 MNR = -5.91
-4.57 MHC = -6.21 0 MNÑ = -0.18MHM = -6.61 MNI = -8.09MHI = -0.98 MÑN = 13.26MIH = 14.63 MÑJ = -7.59
-6.28 MID = -7.10 MÑS = -5.67MIN = -7.90 0MIJ = 0.37
S MJI = 13.95MJE = -6.64 0MJÑ = -7.31
2.90
Ñ
-5.67
-7.59
Ñ
2.90
J
-7.31
-6.64
J
2.90
E
-7.82
+-
MOMENTOS FINALES
0.37 MOK = 0.46 MTO = 4.501.49 0 MOT = 3.85 0 MTU = -4.50 0-1.86 MOP = -4.31 MUT = 13.8917.84 MPO = 16.20 MUP = -6.73 0-7.16 0 MPU = -5.27 0 MUV = -7.16-9.57 MPL = -8.24 MVU = 8.29-1.10 MPQ = -2.70 MVQ = -2.94 013.30 MQP = 11.35 MVW = -5.35-4.84 0 MQV = -2.54 0 MWV = 10.25-6.90 MQM = -5.35 MWR = -4.22 0-1.57 MQR -3.46 MWX = -6.03
14.17 MRQ = 12.35 MXW = 6.02-5.91 0 MRW = -3.63 0 MXS = -6.02 0-0.18 MRS = -2.36-8.09 MRN = -6.3613.26 MSR = 10.86-7.59 0 MSX = -4.57 0-5.67 MSÑ = -6.28
0
0
-
CALCULO DE "C" POR MATRICES
#VALUE!
ECUACION PARA CALCULAR LAS CONSTANTES COEFICIENTESC1 C2 C3
NIVEL = + C1*Q11-C2*Q12-C3*Q13-R1 = 0 #REF! #REF! #REF! 0NIVEL = - C1*Q21+C2*Q22-C3*Q23-R2 = 0 #REF! #REF! #REF! 0NIVEL = - C1*Q31-C2*Q32+C3*Q33-R3 = 0 #REF! #REF! #REF! -
#REF! C1 + #REF! C2 #REF! C3 = 0 MATRIS INVERSA#REF! C + #REF! C2 #REF! C3 = 0 #REF! #REF! #REF!#REF! C1 + #REF! C2 + #REF! C3 = - #REF! #REF! #REF!
#REF! #REF! #REF!
C1 = #VALUE!C2 = #VALUE!C3 = #VALUE!
MOMENTOS FINALES
MAE = #REF! MGF =MBF = #REF! MGK =MCG = #REF! MGH =MDH = #REF! MGC =MEA = #REF! MHG =MEF = #REF! MHL =MEI = MHD =MFE = MIE =MFJ = MIJ =MFG = MIM =MFB =
MJI =MMI = MJN =MMN = MJK =MNM = MJF =MNO = MKJ =MNJ = MKO =MON = MKL =MOP = MKG =MOK = MLK =MPO = MLP =MPL = MLH =
1ᵒ2o3o
C1 = ###C2 = ###C3 = ###
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