DISCIPLINA DE CONCEPÇÃO E DIMENSIONAMENTO DE … · Método dos deslocamentos a uma viga...
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DECivil Departamento de Engenharia Civil e Arquitectura e Georrecursos DISCIPLINA DE CONCEPÇÃO E DIMENSIONAMENTO DE ESTRUTURAS Tabelas Cálculo elástico de vigas
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Transcript of DISCIPLINA DE CONCEPÇÃO E DIMENSIONAMENTO DE … · Método dos deslocamentos a uma viga...
DECivil Departamento de Engenharia Civil e Arquitectura e Georrecursos
DISCIPLINA DE CONCEPÇÃO E DIMENSIONAMENTO DE ESTRUTURAS
Tabelas
Cálculo elástico de vigas
________________________________________________________________________________________________
Francisco VirtuosoDiogo Rego
Março 2014
Tabela de esforços em vigas contínuas
________________________________________________________________________________________________
Tabela de esforços em vigas contínuas - Francisco Virtuoso e Diogo Rego - 2014
________________________________________________________________________________________________
Método dos deslocamentos a uma viga contínua de dois tramos
Método de Cross a uma viga contínua de dois tramos
Cálculo dos momentos positivos e dos esforços transversos numa viga de dois tramos
Cálculo de deslocamentos no vão de um tramo de uma viga continua de dois tramos
24
25
EXEMPLOS DE APLICAÇÃO
22
23
ÍNDICE DAS TABELAS
TABELAS
3
4 - 21
Âmbito
Estas tabelas foram elaboradas para serem utilizadas no âmbito do ensino das disciplinas de Estruturas Metálicas, de Dimensionamento de Estruturas, de
Pontes e de Estruturas Especiais do curso de Mestrado em Engenharia Civil do Instituto Superior Técnico.
ÍNDICE GERAL
_______________________________________________________________________________________________________________________
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N.º de Tramos Tipo de carga N.º da tabela Página
Distribuída 1 2
Concentrada 2 4
Distribuída 3 6
Concentrada 4 8
Distribuída 5 10
Concentrada 6 14
3(L1 ≠ L3)
Distribuída 7 18
TabelaExemplo
ÍNDICE DAS TABELAS
As presentes tabelas estão organizadas de forma a permitir o cálculo da distribuição de esforços, momentos flectores e esforços transversos, em vigas
contínuas devido a cargas distribuídas e concentradas. No caso das cargas distribuídas consideram-se separadamente o caso de cargas uniformes em todo o
comprimento das vigas e os casos correspondentes a cargas variáveis, possibilitando a sua consideração tendo em conta a alternância de sua actuação em
diferentes vãos. Cada tabela é acompanhada de um exemplo da sua aplicação, apresentado numérica e graficamente. Nos exemplos é ilustrado o cálculo de
esforços obtidos por aplicação directa dos valores tabelados e de esforços cujo cálculo requer análises complementares de equilíbrio. Para facilitar, o
documento foi paginado de forma a permitir a consulta simultânea de cada tabela e do exemplo correspondente, pelo que o exemplo é apresentado na página
anterior de acordo com o seguinte esquema:
Modelo
2
3
4
A BL1 L
s1s2
C
p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200
A BL1 L
C
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200
P
A B DL1 L L1
s1s2
s3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 300
C
p
A B D
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 300
C
P
L1 L L1
A B EL1 L L1
s1s2
s4
C
p
L
s3
D
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 31 32 33 34 35 36 37 38 39 400 21 22 23 24 25 26 27 28 29 30
A B E
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 31 32 33 34 35 36 37 38 39 400
C
21 22 23 24 25 26 27 28 29 30
DL1 L L1L
P
A B DαL L
s1s2
s3
C
p
βL
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 300
_______________________________________________________________________________________________________________________
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________________________________________________________________________________________________
n = 8 / 10 = 0.8
p = 20 kN/ms1 = s = 10 kN/m
Momentos Flectoresi)
M1 = +0.036 p L 2 = +72 kNm
MB = -0.105 p L 2 = -210 kNm
M2 = +0.078 p L 2 = +156 kNm
ii)
M1 = +0.063 s L 2 = +63 kNm
MB = -0.036 s L 2 = -36 kNm
i) + ii)
M1 = +135 kNm
MB = -246 kNm
M2 = +152 kNm
Esforços Tranversosi)
VA = +0.269 p L = +54 kN
VBesq = -0.531 p L = -106 kN
VBdir = -0.605 p L = +121 kN
ii)
VA = +0.356 p L = +36 kN
VBesq = -0.444 p L = -44 kN
VBdir = -0.036 p L = +4 kN
i) + ii)
VA = +89 kN
VBesq = -151 kN
VBdir = -125 kN
=
Exemplo (Tabela 1) - Viga contínua de dois tramos com cargas uniformemente distribuídas
+
=
+
A B CL = 81 L = 10
s1
p
10 11 12 13 14 15 16 17 18 19 201 2 3 4 5 6 7 8 90
MB
M1M2
MB
M1
MB
M1
M2
VdirB
VesqB
VA
VA
VesqB
VdirB
VA
VesqB
VdirB
_______________________________________________________________________________________________________________________
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________________________________________________________________________________________________
MB MB MB
p L2 sec p L
2p L
2 sec s1 L2 sec s1 L
2s2 L
2s2 L
2 sec
0.40 0.000 0 -0.095 0.082 16 0.017 5 -0.006 -0.089 0.084 16
0.45 0.000 0 -0.094 0.082 16 0.021 5 -0.008 -0.086 0.086 16
0.50 0.002 1 -0.094 0.082 16 0.026 5 -0.010 -0.083 0.087 16
0.55 0.005 2 -0.094 0.082 16 0.031 5 -0.013 -0.081 0.088 16
0.60 0.010 2 -0.095 0.082 16 0.037 5 -0.017 -0.078 0.089 16
0.65 0.015 3 -0.097 0.081 16 0.042 5 -0.021 -0.076 0.090 16
0.70 0.022 3 -0.099 0.080 16 0.049 4 -0.025 -0.074 0.091 16
0.75 0.029 3 -0.102 0.079 16 0.055 4 -0.030 -0.071 0.091 16
0.80 0.036 3 -0.105 0.078 16 0.063 4 -0.036 -0.069 0.092 16
0.85 0.043 3 -0.109 0.076 16 0.070 4 -0.042 -0.068 0.093 16
0.90 0.052 4 -0.114 0.075 16 0.078 4 -0.048 -0.066 0.094 16
0.95 0.061 4 -0.119 0.072 16 0.086 4 -0.055 -0.064 0.094 16
1.00 0.070 4 -0.125 0.070 16 0.095 4 -0.063 -0.063 0.095 16
1.10 0.090 4 -0.139 0.064 16 0.114 4 -0.079 -0.060 0.096 16
1.20 0.111 4 -0.155 0.059 17 0.134 4 -0.098 -0.057 0.097 16
1.30 0.133 4 -0.174 0.053 17 0.155 4 -0.119 -0.054 0.098 16
VBesq VA VB
esq VBdir VA VB
esq VBdir
p L s1 L s1 L s1 L s2 L s2 L s2 L
0.40 -0.438 0.186 -0.214 0.006 -0.223 0.589 -0.411
0.45 -0.434 0.208 -0.242 0.008 -0.192 0.586 -0.414
0.50 -0.438 0.229 -0.271 0.010 -0.167 0.583 -0.417
0.55 -0.446 0.251 -0.299 0.013 -0.147 0.581 -0.419
0.60 -0.458 0.272 -0.328 0.017 -0.130 0.578 -0.422
0.65 -0.474 0.293 -0.357 0.021 -0.117 0.576 -0.424
0.70 -0.491 0.314 -0.386 0.025 -0.105 0.574 -0.426
0.75 -0.510 0.335 -0.415 0.030 -0.095 0.571 -0.429
0.80 -0.531 0.356 -0.444 0.036 -0.087 0.569 -0.431
0.85 -0.553 0.376 -0.474 0.042 -0.080 0.568 -0.432
0.90 -0.576 0.397 -0.503 0.048 -0.073 0.566 -0.434
0.95 -0.600 0.417 -0.533 0.055 -0.067 0.564 -0.436
1.00 -0.625 0.437 -0.562 0.063 -0.063 0.562 -0.437
1.10 -0.676 0.478 -0.622 0.079 -0.054 0.560 -0.440
1.20 -0.729 0.518 -0.682 0.098 -0.047 0.557 -0.443
1.30 -0.784 0.558 -0.742 0.119 -0.042 0.554 -0.446
0.375
0.605
0.609
0.614
0.619
0.625
0.269
0.297
n = L1 / L
p ≠ 0
Carga Permanente
p L
VBdir
p L
VA
Relaçãodos vãos
p ≠ 0 s1 ≠ 0 e s2 = 0 s1 = 0 e s2 ≠ 0
s1 ≠ 0 e s2 = 0
Sc no 1.º tramo
n = L1 / L
Relaçãodos vãos
Tabela 1 - Viga contínua de dois tramos com cargas uniformemente distribuídas
M1 M2M2 M1
Momentos Flectores
Carga permanente Sc no1.º tramo Sc no 2.º tramo
-0.038
0.016
0.063
0.595
0.594
0.594
Esforço Transverso
s1 = 0 e s2 ≠ 0
Sc no 2.º tramo
0.104
0.424
0.471
0.516
0.324
0.350
0.142
0.176
0.209
0.240
0.639
0.655
0.602
0.674
0.594
0.595
0.597
0.599
A BL1 L
s1s2
C
p
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200
_______________________________________________________________________________________________________________________
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________________________________________________________________________________________________
n = 8 / 10 = 0.8
P = 100 kN
Momentos Flectoresi)
MB = -0.067 P L = -67 kNm
M1 = MB/2 + P L1 / 4 = +167 kNm
ii)
MB = -0.104 P L = -104 kNm
M2 = MB + P L/ 4 = +198 kNm
i) + ii)
MB = -154 kNm
M1 = +115 kNm
M2 = +165 kNm
Esforços Tranversosi)
VA = M1 / 4 = +42 kN
VBesq = (MB - M1) / 4 = -58 kN
VBdir = -MB / 10 = +7 kN
ii)
VBesq = MB / 8 = -13 kN
VBdir = (M2 - MB) / 5 = +60 kN
VC = -M2 / 5 = -40 kN
i) + ii)
VA = +29 kN
VBesq = -71 kN
VBdir = +67 kN
VC = -33 kN
=
+
=
Exemplo (Tabela 2) - Viga contínua de dois tramos com cargas concentradas
+
VA
VesqB
VdirB
VdirB
VesqB
VC
VdirB
VesqB
VC
VA
MB
M2
MB
P
M1
MB
M2
M1
A B
10 11 12 13 14 15 16 17 18 19 20
C
1 2 3 4 5 6 7 8 90
L1 = 8 L = 10
PP
_______________________________________________________________________________________________________________________
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0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.10 1.20 1.30
0 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
1 -0.006 -0.007 -0.008 -0.010 -0.011 -0.013 -0.014 -0.016 -0.018 -0.019 -0.021 -0.023 -0.025 -0.029 -0.032 -0.036
2 -0.011 -0.013 -0.016 -0.019 -0.022 -0.025 -0.028 -0.031 -0.034 -0.037 -0.041 -0.044 -0.048 -0.055 -0.063 -0.071
3 -0.016 -0.019 -0.023 -0.027 -0.031 -0.035 -0.039 -0.044 -0.049 -0.053 -0.058 -0.063 -0.068 -0.079 -0.089 -0.100
4 -0.019 -0.023 -0.028 -0.033 -0.038 -0.043 -0.048 -0.054 -0.060 -0.066 -0.072 -0.078 -0.084 -0.097 -0.110 -0.123
5 -0.021 -0.026 -0.031 -0.037 -0.042 -0.048 -0.054 -0.060 -0.067 -0.073 -0.080 -0.087 -0.094 -0.108 -0.123 -0.138
6 -0.022 -0.027 -0.032 -0.037 -0.043 -0.049 -0.055 -0.062 -0.068 -0.075 -0.082 -0.089 -0.096 -0.111 -0.126 -0.141
7 -0.020 -0.025 -0.030 -0.035 -0.040 -0.046 -0.051 -0.057 -0.063 -0.070 -0.076 -0.083 -0.089 -0.103 -0.117 -0.131
8 -0.016 -0.020 -0.024 -0.028 -0.032 -0.037 -0.042 -0.046 -0.051 -0.056 -0.061 -0.067 -0.072 -0.083 -0.094 -0.106
9 -0.010 -0.012 -0.014 -0.017 -0.019 -0.022 -0.025 -0.027 -0.030 -0.033 -0.036 -0.040 -0.043 -0.049 -0.056 -0.063
10 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
11 -0.061 -0.059 -0.057 -0.055 -0.053 -0.052 -0.050 -0.049 -0.048 -0.046 -0.045 -0.044 -0.043 -0.041 -0.039 -0.037
12 -0.103 -0.099 -0.096 -0.093 -0.090 -0.087 -0.085 -0.082 -0.080 -0.078 -0.076 -0.074 -0.072 -0.069 -0.065 -0.063
13 -0.128 -0.123 -0.119 -0.115 -0.112 -0.108 -0.105 -0.102 -0.099 -0.096 -0.094 -0.092 -0.089 -0.085 -0.081 -0.078
14 -0.137 -0.132 -0.128 -0.124 -0.120 -0.116 -0.113 -0.110 -0.107 -0.104 -0.101 -0.098 -0.096 -0.091 -0.087 -0.083
15 -0.134 -0.129 -0.125 -0.121 -0.117 -0.114 -0.110 -0.107 -0.104 -0.101 -0.099 -0.096 -0.094 -0.089 -0.085 -0.082
16 -0.120 -0.116 -0.112 -0.108 -0.105 -0.102 -0.099 -0.096 -0.093 -0.091 -0.088 -0.086 -0.084 -0.080 -0.076 -0.073
17 -0.098 -0.094 -0.091 -0.088 -0.085 -0.083 -0.080 -0.078 -0.076 -0.074 -0.072 -0.070 -0.068 -0.065 -0.062 -0.059
18 -0.069 -0.066 -0.064 -0.062 -0.060 -0.058 -0.056 -0.055 -0.053 -0.052 -0.051 -0.049 -0.048 -0.046 -0.044 -0.042
19 -0.035 -0.034 -0.033 -0.032 -0.031 -0.030 -0.029 -0.028 -0.028 -0.027 -0.026 -0.025 -0.025 -0.024 -0.023 -0.022
Posi
ção
da c
arga
Tabela 2- Viga contínua de dois tramos com carga pontual
Relação dos vãos: n = L1 / LP L
Momento Flector MBFactor:
A BL1 L
C
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200
P
_______________________________________________________________________________________________________________________
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________________________________________________________________________________________________
n = 8 / 10 = 0.8
p = 20 kN/ms1 = s2 = s = 10 kN/m
Momentos Flectoresi)
M1 = +0.044 p L 2 = +88 kNm
MB = -0.082 p L 2 = -164 kNm
M2 = +0.043 p L 2 = +86 kNm
ii)
M1 = +0.040 s L 2 = +40 kNm
MB = -0.093 s L 2 = -93 kNm
M2 = +0.057 s L 2 = +57 kNm
MC = +0.044 s L 2 = +44 kNm
i) + ii)
M1 = +128 kNm
MB = -257 kNm
M2 = +143 kNm
MC = -120 kNm
Esforços Tranversosi)
VA = +0.297 p L = +59 kN
VBesq = -0.503 p L = -101 kN
VBdir = -0.500 p L = -100 kN
ii)
VA = +0.284 p L = +28 kN
VBesq = -0.516 p L = -52 kN
VBdir = -0.549 p L = -55 kN
VCesq = -0.451 p L = -45 kN
VCdir = -0.055 p L = -6 kN
i) + ii)
VA = +88 kN
VBesq = -152 kN
VBdir = -155 kN
VCesq = -145 kN
VCdir = -106 kN
Exemplo (Tabela 3) - Viga contínua de três tramos com cargas uniformemente distribuídas
+
=
=
+
A B D
10 11 12 13 14 15 16 17 18 19 20
C
p
1 2 3 4 5 6 7 8 90 22 23 24 25 26 27 28 29 3021
L = 81 L = 10 L = 81
s1 s2
MB
M1 M2
MC
MB
M1 M2
MC
MB
M1 M2
VA
VdirB
VesqB
VesqC
VdirC
VdirB
VA
VesqB
VdirB
VA VdirC
VesqB
VesqC
_______________________________________________________________________________________________________________________
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________________________________________________________________________________________________
MB M2 MB MC MB M2 MB MC MB
p L2 sec p L
2p L
2 s1 L2 sec s1 L
2s1 L
2s2 L
2s2 L
2s1 L
2 sec s1 L2
s1 L2 sec s1 L
2s1 L
2 sec s1 L2
0.40 0.000 1 -0.070 0.055 0.017 5 -0.007 0.002 -0.066 0.059 0.000 0 -0.072 0.057 15 -0.063 0.018 5 -0.004
0.45 0.002 2 -0.070 0.055 0.021 5 -0.009 0.003 -0.064 0.061 0.002 1 -0.073 0.058 15 -0.061 0.022 5 -0.006
0.50 0.006 2 -0.070 0.055 0.025 5 -0.012 0.004 -0.063 0.063 0.005 2 -0.074 0.059 15 -0.059 0.027 5 -0.008
0.55 0.010 3 -0.071 0.054 0.030 4 -0.015 0.005 -0.061 0.064 0.009 2 -0.076 0.059 15 -0.056 0.033 5 -0.010
0.60 0.016 3 -0.072 0.053 0.036 4 -0.019 0.006 -0.060 0.065 0.014 3 -0.078 0.059 15 -0.054 0.039 5 -0.013
0.65 0.022 3 -0.074 0.051 0.042 4 -0.023 0.007 -0.058 0.067 0.020 3 -0.081 0.059 15 -0.051 0.045 5 -0.016
0.70 0.029 3 -0.076 0.049 0.048 4 -0.028 0.008 -0.057 0.068 0.026 3 -0.084 0.058 15 -0.049 0.052 5 -0.019
0.75 0.036 4 -0.079 0.046 0.054 4 -0.033 0.009 -0.056 0.069 0.033 3 -0.088 0.058 15 -0.046 0.059 5 -0.023
0.80 0.044 4 -0.082 0.043 0.061 4 -0.039 0.011 -0.054 0.071 0.040 4 -0.093 0.057 15 -0.044 0.066 5 -0.028
0.85 0.052 4 -0.086 0.039 0.069 4 -0.045 0.012 -0.053 0.072 0.048 4 -0.098 0.056 16 -0.041 0.074 5 -0.033
0.90 0.061 4 -0.090 0.035 0.077 4 -0.052 0.014 -0.052 0.073 0.056 4 -0.104 0.055 16 -0.039 0.082 5 -0.038
0.95 0.070 4 -0.095 0.030 0.085 4 -0.059 0.015 -0.051 0.074 0.064 4 -0.110 0.055 16 -0.036 0.091 5 -0.044
1.00 0.080 4 -0.100 0.025 0.093 4 -0.067 0.017 -0.050 0.075 0.073 4 -0.117 0.053 16 -0.033 0.100 4 -0.050
1.10 0.100 4 -0.112 0.013 0.112 4 -0.084 0.020 -0.048 0.077 0.092 4 -0.132 0.050 16 -0.028 0.120 4 -0.064
1.20 0.122 4 -0.126 -0.001 0.131 4 -0.104 0.024 -0.046 0.079 0.113 4 -0.150 0.046 16 -0.023 0.141 4 -0.080
1.30 0.146 4 -0.143 -0.018 0.153 4 -0.125 0.027 -0.045 0.080 0.135 4 -0.170 0.042 17 -0.017 0.164 4 -0.098
VA VBdir VA VB
esq VBdir VC
dir VA VBdir VA VB
dir VCesq VC
dir VA
p L p L s1 L s1 L s1 L s1 L s2 L s2 L s1 L s1 L s1 L s1 L s1 L
0.40 0.025 0.500 0.184 -0.216 0.009 -0.006 -0.164 0.500 0.019 0.509 -0.491 0.159 0.189
0.45 0.070 0.500 0.205 -0.245 0.012 -0.007 -0.142 0.500 0.063 0.512 -0.488 0.136 0.212
0.50 0.109 0.500 0.227 -0.273 0.016 -0.008 -0.125 0.500 0.102 0.516 -0.484 0.117 0.234
0.55 0.146 0.500 0.248 -0.302 0.020 -0.009 -0.111 0.500 0.137 0.520 -0.480 0.102 0.257
0.60 0.179 0.500 0.269 -0.331 0.025 -0.010 -0.099 0.500 0.170 0.525 -0.475 0.089 0.279
0.65 0.211 0.500 0.290 -0.360 0.030 -0.011 -0.089 0.500 0.200 0.530 -0.470 0.079 0.300
0.70 0.241 0.500 0.311 -0.389 0.036 -0.012 -0.081 0.500 0.229 0.536 -0.464 0.070 0.322
0.75 0.270 0.500 0.331 -0.419 0.042 -0.013 -0.074 0.500 0.257 0.542 -0.458 0.062 0.344
0.80 0.297 0.500 0.352 -0.448 0.049 -0.013 -0.068 0.500 0.284 0.549 -0.451 0.055 0.365
0.85 0.324 0.500 0.372 -0.478 0.057 -0.014 -0.063 0.500 0.310 0.557 -0.443 0.048 0.387
0.90 0.350 0.500 0.393 -0.507 0.065 -0.015 -0.058 0.500 0.335 0.565 -0.435 0.043 0.408
0.95 0.375 0.500 0.413 -0.537 0.074 -0.016 -0.054 0.500 0.359 0.574 -0.426 0.038 0.429
1.00 0.400 0.500 0.433 -0.567 0.083 -0.017 -0.050 0.500 0.383 0.583 -0.417 0.033 0.450
1.10 0.448 0.500 0.474 -0.626 0.104 -0.018 -0.044 0.500 0.430 0.604 -0.396 0.026 0.492
1.20 0.495 0.500 0.514 -0.686 0.127 -0.020 -0.039 0.500 0.475 0.627 -0.373 0.019 0.533
1.30 0.540 0.500 0.554 -0.746 0.153 -0.021 -0.034 0.500 0.519 0.653 -0.347 0.013 0.575
Momentos Flectores
Sc no 1.º tramo
M1
Sc no 1.º e 3.º tramos
s1 = s2 ≠ 0 e s3 = 0
Tabela 3 - Viga contínua de três tramos com cargas uniformemente distribuídas
p ≠ 0
Carga permanente
M1M1
Sc no 1.º e 2.º tramosSc no 2.º tramo
n = L1 / L
s1 ≠ 0 e s2 = s3 = 0
Relaçãodos vãos
s1 = s2 ≠ 0 e s3 = 0
Sc no 1.º e 2.º tramos
s2 ≠ 0 e s1 = s3 = 0
Esforço TransversoSc no 1.º e 3.º tramos
s1 = s3 ≠ 0 e s2 = 0
s1 = s3 ≠ 0 e s2 = 0
M1 M2
Sc no 1.º tramos1 ≠ 0 e s2 = s3 = 0
p L
Sc no 2.º tramos2 ≠ 0 e s1 = s3 = 0
-0.375
p ≠ 0
Carga Permanente
VBesq
-0.450
Relaçãodos vãos
n = L1 / L
-0.705
-0.526
-0.550
-0.575
-0.600
-0.439
-0.459
VBesq
s1 L
-0.381
-0.387
-0.652
-0.380
-0.391
-0.404
-0.421
-0.503
-0.480
VBesq
s1 L
-0.321
-0.350
-0.670
-0.725
-0.540
-0.565
-0.591
-0.617 -0.550
-0.471
-0.266
-0.293
-0.378
-0.493
-0.516
-0.398
-0.413
-0.430
-0.406
-0.211
-0.238
-0.608
-0.667
-0.435
-0.463
-0.492
-0.521
-0.725-0.781-0.760
A B DL1 L L1
s1s2
s3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 300
C
p
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n = 8 / 10 = 0.8
P = 100 kN
Momentos Flectoresi)
MB = -0.072 P L = -72 kNm
MC = -0.020 P L = +20 kNm
M1 = MB/2 + P L1 /4 = +164 kNm
ii)
MB = -0.082 P L = -82 kNm
M2 = MB + P L/ 4 = +168 kNm
i) + ii)
MB = -154 kNm
MC = -62 kNm
M1 = +123 kNm
M2 = +142 kNm
Esforços Tranversosi)
VA = M1 / 4 = +41 kN
VBesq = (MB - M1) / 4 = -59 kN
VBdir = (Mc -MB) / 10 = +9 kN
VCdir = -MC / 8 = -3 kN
ii)
VBesq = MB / 8 = -10 kN
VBdir = (M2 - MB) / 5 = +50 kN
VCesq = (MC - M2) / 5 = -50 kN
VCdir = -MC / 8 = +10 kN
i) + ii)
VA = +31 kN
VBesq = -69 kN
VBdir = +59 kN
VCesq = -41 kN
VCdir = +8 kN
Exemplo (Tabela 4) - Viga contínua de três tramos com cargas concentradas
=
+
=
+
B D
10 11 12 13 14 15 16 17 18 19 20
C
3 4 5 6 7 8 9 22 23 24 25 26 27 28 29 3021
A
1 20
L = 81 L = 10 L = 81
PP
VdirB
VesqB
VesqC
VdirC
VesqB
VdirB
VA
VesqC
VdirC
VA
VesqB
VdirB
VdirC
MC
M1
MB
M2
MB
MB
M2M1
MC
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0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.10 1.20 1.30
0 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
1 -0.006 -0.008 -0.009 -0.011 -0.012 -0.014 -0.016 -0.017 -0.019 -0.021 -0.023 -0.025 -0.026 -0.030 -0.034 -0.038
2 -0.013 -0.015 -0.018 -0.021 -0.024 -0.027 -0.030 -0.034 -0.037 -0.040 -0.044 -0.048 -0.051 -0.059 -0.066 -0.074
3 -0.018 -0.022 -0.026 -0.030 -0.034 -0.038 -0.043 -0.048 -0.053 -0.058 -0.063 -0.068 -0.073 -0.083 -0.094 -0.105
4 -0.022 -0.027 -0.032 -0.037 -0.042 -0.047 -0.053 -0.059 -0.065 -0.071 -0.077 -0.083 -0.090 -0.103 -0.116 -0.130
5 -0.025 -0.030 -0.035 -0.041 -0.047 -0.053 -0.059 -0.066 -0.072 -0.079 -0.086 -0.093 -0.100 -0.115 -0.129 -0.145
6 -0.025 -0.030 -0.036 -0.042 -0.048 -0.054 -0.061 -0.067 -0.074 -0.081 -0.088 -0.095 -0.102 -0.117 -0.133 -0.148
7 -0.023 -0.028 -0.033 -0.039 -0.045 -0.050 -0.056 -0.062 -0.069 -0.075 -0.082 -0.088 -0.095 -0.109 -0.123 -0.138
8 -0.019 -0.023 -0.027 -0.031 -0.036 -0.041 -0.045 -0.050 -0.055 -0.061 -0.066 -0.071 -0.077 -0.088 -0.099 -0.111
9 -0.011 -0.014 -0.016 -0.019 -0.021 -0.024 -0.027 -0.030 -0.033 -0.036 -0.039 -0.042 -0.046 -0.052 -0.059 -0.066
10 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
11 -0.056 -0.054 -0.052 -0.050 -0.049 -0.047 -0.046 -0.044 -0.043 -0.042 -0.041 -0.040 -0.039 -0.037 -0.036 -0.034
12 -0.090 -0.087 -0.084 -0.081 -0.079 -0.077 -0.075 -0.073 -0.071 -0.069 -0.067 -0.066 -0.064 -0.061 -0.059 -0.056
13 -0.106 -0.103 -0.100 -0.097 -0.094 -0.092 -0.089 -0.087 -0.085 -0.083 -0.081 -0.079 -0.077 -0.074 -0.071 -0.068
14 -0.108 -0.105 -0.102 -0.099 -0.097 -0.094 -0.092 -0.090 -0.087 -0.085 -0.084 -0.082 -0.080 -0.077 -0.074 -0.071
15 -0.099 -0.096 -0.094 -0.091 -0.089 -0.087 -0.085 -0.083 -0.082 -0.080 -0.078 -0.077 -0.075 -0.072 -0.069 -0.067
16 -0.081 -0.080 -0.078 -0.076 -0.075 -0.073 -0.072 -0.070 -0.069 -0.068 -0.066 -0.065 -0.064 -0.062 -0.060 -0.058
17 -0.060 -0.059 -0.058 -0.057 -0.056 -0.055 -0.054 -0.053 -0.052 -0.051 -0.051 -0.050 -0.049 -0.047 -0.046 -0.045
18 -0.036 -0.036 -0.036 -0.036 -0.035 -0.035 -0.035 -0.034 -0.034 -0.033 -0.033 -0.032 -0.032 -0.031 -0.030 -0.030
19 -0.016 -0.016 -0.016 -0.016 -0.016 -0.016 -0.016 -0.016 -0.016 -0.015 -0.015 -0.015 -0.015 -0.015 -0.014 -0.014
20 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
21 0.004 0.005 0.005 0.006 0.007 0.007 0.008 0.009 0.009 0.010 0.010 0.011 0.011 0.012 0.013 0.014
22 0.007 0.008 0.009 0.010 0.011 0.012 0.013 0.014 0.015 0.016 0.017 0.018 0.019 0.021 0.023 0.024
23 0.008 0.010 0.011 0.013 0.014 0.015 0.017 0.018 0.019 0.020 0.022 0.023 0.024 0.026 0.028 0.030
24 0.009 0.011 0.012 0.013 0.015 0.016 0.018 0.019 0.021 0.022 0.023 0.024 0.026 0.028 0.030 0.032
25 0.009 0.010 0.012 0.013 0.015 0.016 0.017 0.019 0.020 0.021 0.023 0.024 0.025 0.027 0.029 0.031
26 0.008 0.009 0.011 0.012 0.013 0.014 0.016 0.017 0.018 0.019 0.020 0.021 0.022 0.024 0.026 0.028
27 0.006 0.007 0.009 0.010 0.011 0.012 0.013 0.014 0.015 0.016 0.016 0.017 0.018 0.020 0.021 0.023
28 0.004 0.005 0.006 0.007 0.007 0.008 0.009 0.010 0.010 0.011 0.012 0.012 0.013 0.014 0.015 0.016
29 0.002 0.003 0.003 0.003 0.004 0.004 0.005 0.005 0.005 0.006 0.006 0.006 0.007 0.007 0.008 0.008
Posi
ção
da c
arga
Tabela 4 - Viga contínua de três tramos com carga pontual (P)
Relação dos vãos: n = L1 / LP L
Momento Flector MBFactor:
A B D
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 300
C
P
L1 L L1
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n = 8 / 10 = 0.8
p = 20 kN/ms1 = s3 = s = 10 kN/m
Momentos Flectoresi)
MB = -0.082 p L 2 = -164 kNm
MC = +0.084 s L 2 = -168 kNm
MD = +0.082 p L 2 = -164 kNm
M1 = VA x 3/2 = +89 kNm
M2 = (MB+MC)/2 + pL2/8 = +84 kNm
ii)
M1 = +0.069 s L 2 = +69 kNm
M2 = +0.075 s L 2 = +75 kNm
i) + ii)
M1 = +158 kNm
M2 = +159 kNm
Esforços Tranversosi)
VA = +0.298 p L = +60 kN
VBdir = -0.497 p L = +99 kN
VCdir = -0.503 p L = +101 kN
VDdir = -0.502 p L = +100 kN
ii)
VA = +0.370 p L = +37 kN
VBdir = -0.018 p L = -2 kN
VCdir = +0.484 p L = +48 kN
VDdir = +0.072 p L = +7 kN
i) + ii)
VA = +97 kNm
VBdir = +98 kNm
VCdir = +149 kNm
VDdir = +108 kNm
Exemplo (Tabela 5 - Folha 1) - Viga contínua de quatro tramos com cargas uniformemente distribuídas
+
=
=
+
A B D
s1 s3
10 11 12 13 14 15 16 17 18 19 20
C
1 2 3 4 5 6 7 8 90
L = 81 L = 10 L = 81
E
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
p
L = 10
VA
VdirB Vdir
C VdirD
VAVdir
B
VdirC
VdirD
VA
VdirB
VdirC
VdirD
MB MC MD
M1 M3
M1 M3
M1 M3
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MB MD
p L2
p L2 s1 L
2 sec s1 L2 sec s2 L
2 sec s2 L2 sec
0.40 -0.061 -0.061 0.026 4 0.065 25 0.065 15 0.026 34
0.45 -0.062 -0.062 0.030 4 0.067 25 0.067 15 0.030 35
0.50 -0.063 -0.063 0.034 4 0.068 25 0.068 15 0.034 35
0.55 -0.064 -0.064 0.039 4 0.069 25 0.069 15 0.039 35
0.60 -0.066 -0.066 0.044 4 0.070 25 0.070 15 0.044 35
0.65 -0.069 -0.069 0.049 4 0.072 25 0.072 15 0.049 35
0.70 -0.073 -0.073 0.055 4 0.073 25 0.073 15 0.055 35
0.75 -0.077 -0.077 0.062 4 0.074 25 0.074 15 0.062 35
0.80 -0.082 -0.082 0.069 4 0.075 25 0.075 15 0.069 35
0.85 -0.087 -0.087 0.076 4 0.076 25 0.076 15 0.076 35
0.90 -0.093 -0.093 0.083 4 0.078 25 0.078 15 0.083 35
0.95 -0.100 -0.100 0.091 4 0.079 25 0.079 15 0.091 36
1.00 -0.107 -0.107 0.100 4 0.081 25 0.081 15 0.100 36
1.10 -0.124 -0.124 0.118 4 0.083 25 0.083 15 0.118 36
1.20 -0.143 -0.143 0.137 4 0.087 25 0.087 15 0.137 36
1.30 -0.164 -0.164 0.158 4 0.090 25 0.090 15 0.158 36
VA VBdir VC
dir VDdir VA VB
dir VCdir VD
dir VA VBdir VC
dir VDdir
p L p L p L p L s1 L s1 L s1 L s1 L s2 L s2 L s2 L s2 L
0.40 0.047 0.467 0.533 0.353 0.228 -0.058 0.475 0.181 -0.181 0.525 0.058 0.172
0.45 0.088 0.467 0.533 0.362 0.244 -0.055 0.477 0.155 -0.155 0.523 0.055 0.206
0.50 0.125 0.469 0.531 0.375 0.260 -0.052 0.479 0.135 -0.135 0.521 0.052 0.240
0.55 0.159 0.471 0.529 0.391 0.278 -0.048 0.481 0.119 -0.119 0.519 0.048 0.272
0.60 0.190 0.474 0.526 0.410 0.296 -0.043 0.482 0.106 -0.106 0.518 0.043 0.304
0.65 0.219 0.479 0.521 0.431 0.314 -0.038 0.483 0.095 -0.095 0.517 0.038 0.336
0.70 0.246 0.484 0.516 0.454 0.333 -0.032 0.484 0.086 -0.086 0.516 0.032 0.367
0.75 0.273 0.490 0.510 0.477 0.351 -0.026 0.484 0.079 -0.079 0.516 0.026 0.399
0.80 0.298 0.497 0.503 0.502 0.370 -0.018 0.484 0.072 -0.072 0.516 0.018 0.430
0.85 0.323 0.506 0.494 0.527 0.389 -0.010 0.484 0.067 -0.067 0.516 0.010 0.461
0.90 0.347 0.515 0.485 0.553 0.408 -0.002 0.484 0.062 -0.062 0.516 0.002 0.492
0.95 0.370 0.525 0.475 0.580 0.427 0.008 0.483 0.057 -0.057 0.517 -0.008 0.523
1.00 0.393 0.536 0.464 0.607 0.446 0.018 0.482 0.054 -0.054 0.518 -0.018 0.554
1.10 0.438 0.561 0.439 0.662 0.485 0.040 0.480 0.047 -0.047 0.520 -0.040 0.615
1.20 0.481 0.589 0.411 0.719 0.523 0.065 0.476 0.042 -0.042 0.524 -0.065 0.677
1.30 0.523 0.622 0.378 0.777 0.562 0.093 0.472 0.038 -0.038 0.528 -0.093 0.738
n =L1 / L
Carga Permanente Sc no 1.º e 3.º tramo Relaçãodos vãos p ≠ 0
-0.094
s1 = s3 ≠ 0 e s2 = s4 = 0
M1 M3MC
p L2
-0.094
-0.094
Sc no 2.º e 4.º tramo
s2 = s4 ≠ 0 e s1 = s3 = 0
M2 M4
-0.087
-0.084
-0.081
-0.078
-0.093
-0.092
-0.090
-0.089
Carga Permanente
p ≠ 0
-0.075
-0.071
-0.063
-0.054
Tabela 5 (Folha 1/2) - Viga contínua de quatro tramos com cargas uniformemente distribuídas
Momentos Flectores
Esforços Transversos
n =L1 / L
Sc no 1.º e 3.º tramo
s1 = s3 ≠ 0 e s2 = s4 = 0
Sc no 2.º e 4.º tramo
s2 = s4 ≠ 0 e s1 = s3 = 0
-0.043
Relaçãodos vãos
A B EL1 L L1
s1s2
s4
C
p
L
s3
D
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 31 32 33 34 35 36 37 38 39 400 21 22 23 24 25 26 27 28 29 30
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n = 8 / 10 = 0.8s2 = s3 = s = 10 kN/m
Momentos Flectoresi)
MB = -0.055 p L 2 = -55 kNm
MC = -0.052 s L 2 = -52 kNm
MD = +0.015 p L 2 = +15 kNm
ii)
MB = +0.015 p L 2 = +15 kNm
MC = -0.052 s L 2 = -52 kNm
MD = +0.055 p L 2 = -55 kNm
i) + ii)
MB = -40 kNm
MC = -104 kNm
MD = +40 kNm
Esforços Tranversosi)
VA = -0.069 p L = -7 kN
VBdir = +0.502 p L = +50 kN
VCdir = +0.067 p L = +7 kN
VDdir = -0.018 p L = -2 kN
ii)
VA = +0.018 p L = +2 kN
VBdir = -0.067 p L = -7 kN
VCdir = +0.498 p L = +50 kN
VDdir = +0.069 p L = +7 kN
i) + ii)
VA = -5 kNm
VBdir = -44 kNm
VCdir = +57 kNm
VDdir = +5 kNm
Exemplo (Tabela 5 - Folha 2) - Viga contínua de quatro tramos com cargas uniformemente distribuídas
+
=
=
+
VA
VdirB
VdirC Vdir
D
VA
VdirB
VdirC
VdirD
VA
VdirB
VdirC
VdirD
2
A B D
s3
10 11 12 13 14 15 16 17 18 19 20
C
1 2 3 4 5 6 7 8 90
L = 81 L = 10 L = 81
E
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
L = 10
s
MB MC
MD
MD
MB
MC
MB
MC
MD
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MB MD MB MD MB MD MB MD
s1 L2
s1 L2
s2 L2
s2 L2
s3 L2
s3 L2
s4 L2
s4 L2
0.40 -0.006 -0.001 -0.072 0.017 0.017 -0.072 -0.001 -0.006
0.45 -0.009 -0.001 -0.069 0.017 0.017 -0.069 -0.001 -0.009
0.50 -0.011 -0.001 -0.067 0.017 0.017 -0.067 -0.001 -0.011
0.55 -0.015 -0.001 -0.064 0.016 0.016 -0.064 -0.001 -0.015
0.60 -0.018 -0.002 -0.062 0.016 0.016 -0.062 -0.002 -0.018
0.65 -0.023 -0.002 -0.060 0.016 0.016 -0.060 -0.002 -0.023
0.70 -0.027 -0.002 -0.058 0.015 0.015 -0.058 -0.002 -0.027
0.75 -0.033 -0.003 -0.057 0.015 0.015 -0.057 -0.003 -0.033
0.80 -0.038 -0.003 -0.055 0.015 0.015 -0.055 -0.003 -0.038
0.85 -0.045 -0.003 -0.053 0.014 0.014 -0.053 -0.003 -0.045
0.90 -0.052 -0.004 -0.052 0.014 0.014 -0.052 -0.004 -0.052
0.95 -0.059 -0.004 -0.050 0.014 0.014 -0.050 -0.004 -0.059
1.00 -0.067 -0.004 -0.049 0.013 0.013 -0.049 -0.004 -0.067
1.10 -0.085 -0.005 -0.047 0.013 0.013 -0.047 -0.005 -0.085
1.20 -0.104 -0.006 -0.044 0.012 0.012 -0.044 -0.006 -0.104
1.30 -0.127 -0.007 -0.042 0.012 0.012 -0.042 -0.007 -0.127
VA VBdir VC
dir VDdir VA VB
dir VCdir VD
dir VA VBdir VC
dir VDdir VA VB
dir VCdir VD
dir
s1 L s1 L s1 L s1 L s2 L s2 L s2 L s2 L s3 L s3 L s3 L s3 L s4 L s4 L s4 L s4 L
0.40 0.184 0.008 -0.002 0.002 -0.180 0.523 0.066 -0.044 0.044 -0.066 0.477 0.180 -0.002 0.002 -0.008 0.216
0.45 0.206 0.011 -0.003 0.002 -0.154 0.520 0.067 -0.038 0.038 -0.067 0.480 0.154 -0.002 0.003 -0.011 0.244
0.50 0.227 0.015 -0.004 0.002 -0.133 0.517 0.067 -0.033 0.033 -0.067 0.483 0.133 -0.002 0.004 -0.015 0.273
0.55 0.248 0.019 -0.005 0.002 -0.117 0.514 0.067 -0.030 0.030 -0.067 0.486 0.117 -0.002 0.005 -0.019 0.302
0.60 0.269 0.023 -0.007 0.003 -0.104 0.511 0.067 -0.027 0.027 -0.067 0.489 0.104 -0.003 0.007 -0.023 0.331
0.65 0.290 0.029 -0.008 0.003 -0.093 0.509 0.067 -0.024 0.024 -0.067 0.491 0.093 -0.003 0.008 -0.029 0.360
0.70 0.311 0.035 -0.010 0.003 -0.083 0.507 0.067 -0.022 0.022 -0.067 0.493 0.083 -0.003 0.010 -0.035 0.389
0.75 0.331 0.041 -0.011 0.003 -0.075 0.504 0.067 -0.020 0.020 -0.067 0.496 0.075 -0.003 0.011 -0.041 0.419
0.80 0.352 0.049 -0.013 0.004 -0.069 0.502 0.067 -0.018 0.018 -0.067 0.498 0.069 -0.004 0.013 -0.049 0.448
0.85 0.372 0.057 -0.015 0.004 -0.063 0.501 0.067 -0.017 0.017 -0.067 0.499 0.063 -0.004 0.015 -0.057 0.478
0.90 0.393 0.065 -0.017 0.004 -0.058 0.499 0.067 -0.016 0.016 -0.067 0.501 0.058 -0.004 0.017 -0.065 0.507
0.95 0.413 0.075 -0.020 0.004 -0.053 0.497 0.067 -0.014 0.014 -0.067 0.503 0.053 -0.004 0.020 -0.075 0.537
1.00 0.433 0.085 -0.022 0.004 -0.049 0.496 0.067 -0.013 0.013 -0.067 0.504 0.049 -0.004 0.022 -0.085 0.567
1.10 0.473 0.107 -0.028 0.005 -0.042 0.493 0.067 -0.012 0.012 -0.067 0.507 0.042 -0.005 0.028 -0.107 0.627
1.20 0.513 0.132 -0.034 0.005 -0.037 0.490 0.067 -0.010 0.010 -0.067 0.510 0.037 -0.005 0.034 -0.132 0.687
1.30 0.553 0.160 -0.041 0.006 -0.033 0.488 0.067 -0.009 0.009 -0.067 0.512 0.033 -0.006 0.041 -0.160 0.747
n =L1 / LMC
s1 L2
MC
s2 L2
Sc no 2.º tramo Sc no 3.º tramo Sc no 4.º tramoRelaçãodos vãos s2 ≠ 0 e s1 = s3 = s4 = 0s1 ≠ 0 e s2 = s3 = s4 = 0
Sc no 1.º tramo
s3 ≠ 0 e s1 = s2 = s4 = 0 s4 ≠ 0 e s1 = s2 = s3 = 0
0.005
0.006
0.007
0.009
0.002
0.002
0.003
0.004
0.018
0.022
0.028
0.033
0.010
0.012
0.014
0.016
-0.051
-0.051
-0.052
-0.052
-0.049
-0.049
-0.050
-0.050
-0.054
-0.054
-0.055
-0.052
-0.053
-0.053
-0.053
-0.050
-0.050
-0.051
-0.051
MC
s3 L2
-0.049
-0.049
0.010
0.012
0.014
0.016
-0.054
-0.054
-0.052
-0.052
-0.052
-0.053
-0.053
-0.053
MC
s4 L2
0.002
0.002
0.003
0.004
0.007
0.009
0.005
0.006
0.022
-0.054
Relaçãodos vãos
-0.054
-0.055 0.033
Sc no 1.º tramo Sc no 2.º tramo Sc no 3.º tramo Sc no 4.º tramo
n =L1 / L
Tabela 5 (Folha 2/2) - Viga contínua de quatro tramos com cargas uniformemente distribuídas
Momentos Flectores
Esforços Transversos
s1 ≠ 0 e s2 = s3 = s4 = 0 s2 ≠ 0 e s1 = s3 = s4 = 0 s3 ≠ 0 e s1 = s2 = s4 = 0 s4 ≠ 0 e s1 = s2 = s3 = 0
0.028
0.018
A B EL1 L L1
s1s2
s4
CL
s3
D
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 31 32 33 34 35 36 37 38 39 400 21 22 23 24 25 26 27 28 29 30
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n = 8 / 10 = 0.8
P = 100 kN
Momentos Flectoresi)
MB = -0.082 P L = -82 kNm
MC = -0.079 P L = -79 kNm
MD = +0.022 P L = +22 kNm
M2 = (MB+MC)/2 + PL/4 = +170 kNm
ii)
MB = +0.022 P L = +22 kNm
MC = -0.079 P L = -79 kNm
MD = +0.082 P L = -82 kNm
M3 = (MC+MD)/2 + PL/4 = +170 kNm
i) + ii)
MB = -60 kNm
MC = -158 kNm
MD = -60 kNm
M2 = +141 kNm
M3 = +141 kNm
Esforços Tranversosi)
VBesq = MB / 8 = -10 kN
VBdir = (M2 - MB) / 5 = +50 kN
VCesq = (MC - M2) / 5 = -50 kN
VCdir = (MD - MC) / 10 = +10 kN
ii)
VBesq = MB / 8 = +3 kN
VBdir = (MC - MB) / 10 = -10 kN
VCesq = (MC - MB) / 10 = -10 kN
VCdir = (M3 - MC) / 5 = +50 kN
i) + ii)
VBesq = -8 kN
VBdir = +40 kN
VCesq = -60 kN
VCdir = +60 kN
Exemplo (Tabela 6) - Viga contínua de quatro tramos com cargas concentradas
+
=
=
+
MC
MD
MB
M2
MB
MC MD
M3
M3
MB
MC
MD
M2
VdirB
VesqB
VesqC
VdirC
A B DL = 10
P P
10 11 12 13 14 15 16 17 18 19 20
C
1 2 3 4 5 6 7 8 90
L = 81 L = 10 L = 81
E
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
VdirB
VesqB
VesqC
VdirC
VdirC
VesqC
VdirB
VesqB
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0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.10 1.20 1.30
0 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
1 -0.006 -0.008 -0.009 -0.011 -0.012 -0.014 -0.016 -0.017 -0.019 -0.021 -0.023 -0.025 -0.027 -0.030 -0.034 -0.039
2 -0.012 -0.015 -0.018 -0.021 -0.024 -0.027 -0.030 -0.033 -0.037 -0.040 -0.044 -0.048 -0.051 -0.059 -0.067 -0.075
3 -0.017 -0.021 -0.025 -0.029 -0.034 -0.038 -0.043 -0.048 -0.052 -0.057 -0.063 -0.068 -0.073 -0.084 -0.095 -0.106
4 -0.021 -0.026 -0.031 -0.036 -0.041 -0.047 -0.053 -0.059 -0.065 -0.071 -0.077 -0.083 -0.090 -0.103 -0.117 -0.131
5 -0.024 -0.029 -0.034 -0.040 -0.046 -0.052 -0.059 -0.065 -0.072 -0.079 -0.086 -0.093 -0.100 -0.115 -0.131 -0.146
6 -0.024 -0.030 -0.035 -0.041 -0.047 -0.054 -0.060 -0.067 -0.074 -0.081 -0.088 -0.095 -0.103 -0.118 -0.134 -0.150
7 -0.023 -0.028 -0.033 -0.038 -0.044 -0.050 -0.056 -0.062 -0.069 -0.075 -0.082 -0.089 -0.096 -0.110 -0.124 -0.139
8 -0.018 -0.022 -0.026 -0.031 -0.035 -0.040 -0.045 -0.050 -0.055 -0.061 -0.066 -0.072 -0.077 -0.089 -0.100 -0.112
9 -0.011 -0.013 -0.016 -0.018 -0.021 -0.024 -0.027 -0.030 -0.033 -0.036 -0.039 -0.042 -0.046 -0.053 -0.060 -0.067
10 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
11 -0.057 -0.055 -0.053 -0.051 -0.049 -0.048 -0.046 -0.045 -0.043 -0.042 -0.041 -0.040 -0.039 -0.037 -0.035 -0.033
12 -0.093 -0.090 -0.086 -0.083 -0.081 -0.078 -0.075 -0.073 -0.071 -0.069 -0.067 -0.065 -0.063 -0.060 -0.057 -0.055
13 -0.112 -0.108 -0.104 -0.100 -0.097 -0.093 -0.091 -0.088 -0.085 -0.083 -0.080 -0.078 -0.076 -0.072 -0.069 -0.066
14 -0.116 -0.111 -0.107 -0.104 -0.100 -0.097 -0.094 -0.091 -0.088 -0.086 -0.083 -0.081 -0.079 -0.075 -0.071 -0.068
15 -0.108 -0.104 -0.100 -0.097 -0.093 -0.090 -0.087 -0.085 -0.082 -0.080 -0.078 -0.076 -0.074 -0.070 -0.067 -0.064
16 -0.091 -0.088 -0.085 -0.082 -0.079 -0.077 -0.074 -0.072 -0.070 -0.068 -0.066 -0.064 -0.063 -0.059 -0.057 -0.054
17 -0.069 -0.067 -0.064 -0.062 -0.060 -0.058 -0.056 -0.055 -0.053 -0.052 -0.050 -0.049 -0.048 -0.045 -0.043 -0.041
18 -0.045 -0.043 -0.042 -0.040 -0.039 -0.038 -0.037 -0.035 -0.034 -0.033 -0.033 -0.032 -0.031 -0.029 -0.028 -0.027
19 -0.021 -0.020 -0.019 -0.019 -0.018 -0.017 -0.017 -0.016 -0.016 -0.015 -0.015 -0.015 -0.014 -0.014 -0.013 -0.012
20 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
21 0.015 0.014 0.014 0.013 0.013 0.013 0.012 0.012 0.012 0.011 0.011 0.011 0.010 0.010 0.010 0.009
22 0.024 0.023 0.022 0.022 0.021 0.021 0.020 0.019 0.019 0.018 0.018 0.018 0.017 0.016 0.016 0.015
23 0.028 0.027 0.027 0.026 0.025 0.024 0.024 0.023 0.023 0.022 0.022 0.021 0.021 0.020 0.019 0.018
24 0.029 0.028 0.027 0.027 0.026 0.025 0.025 0.024 0.023 0.023 0.022 0.022 0.021 0.021 0.020 0.019
25 0.026 0.026 0.025 0.024 0.024 0.023 0.023 0.022 0.022 0.021 0.021 0.021 0.020 0.019 0.019 0.018
26 0.022 0.021 0.021 0.020 0.020 0.020 0.019 0.019 0.018 0.018 0.018 0.017 0.017 0.017 0.016 0.015
27 0.016 0.016 0.015 0.015 0.015 0.015 0.014 0.014 0.014 0.014 0.014 0.013 0.013 0.013 0.012 0.012
28 0.010 0.010 0.010 0.010 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.008 0.008 0.008
29 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004
30 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
31 -0.001 -0.001 -0.001 -0.002 -0.002 -0.002 -0.002 -0.002 -0.002 -0.003 -0.003 -0.003 -0.003 -0.003 -0.004 -0.004
32 -0.002 -0.002 -0.002 -0.003 -0.003 -0.003 -0.004 -0.004 -0.004 -0.004 -0.005 -0.005 -0.005 -0.006 -0.006 -0.006
33 -0.002 -0.003 -0.003 -0.003 -0.004 -0.004 -0.004 -0.005 -0.005 -0.005 -0.006 -0.006 -0.006 -0.007 -0.007 -0.008
34 -0.002 -0.003 -0.003 -0.004 -0.004 -0.004 -0.005 -0.005 -0.006 -0.006 -0.006 -0.007 -0.007 -0.007 -0.008 -0.009
35 -0.002 -0.003 -0.003 -0.004 -0.004 -0.004 -0.005 -0.005 -0.005 -0.006 -0.006 -0.006 -0.007 -0.007 -0.008 -0.008
36 -0.002 -0.002 -0.003 -0.003 -0.004 -0.004 -0.004 -0.005 -0.005 -0.005 -0.005 -0.006 -0.006 -0.007 -0.007 -0.008
37 -0.002 -0.002 -0.002 -0.003 -0.003 -0.003 -0.003 -0.004 -0.004 -0.004 -0.004 -0.005 -0.005 -0.005 -0.006 -0.006
38 -0.001 -0.001 -0.002 -0.002 -0.002 -0.002 -0.002 -0.003 -0.003 -0.003 -0.003 -0.003 -0.003 -0.004 -0.004 -0.004
39 -0.001 -0.001 -0.001 -0.001 -0.001 -0.001 -0.001 -0.001 -0.001 -0.002 -0.002 -0.002 -0.002 -0.002 -0.002 -0.002
Posi
ção
da c
arga
Tabela 6 (Folha 1/3) - Viga contínua de quatro tramos com carga pontual
Relação dos vãos: n = L1 / LP L
Momento Flector MBFactor:
A B E
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 31 32 33 34 35 36 37 38 39 400
C
21 22 23 24 25 26 27 28 29 30
DL1 L L1L
P
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________________________________________________________________________________________________
0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.10 1.20 1.30
0 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
1 0.002 0.002 0.002 0.003 0.003 0.004 0.004 0.005 0.005 0.006 0.006 0.007 0.007 0.008 0.009 0.010
2 0.003 0.004 0.005 0.006 0.006 0.007 0.008 0.009 0.010 0.011 0.012 0.013 0.014 0.016 0.018 0.020
3 0.005 0.006 0.007 0.008 0.009 0.010 0.012 0.013 0.014 0.015 0.017 0.018 0.020 0.022 0.025 0.028
4 0.006 0.007 0.008 0.010 0.011 0.013 0.014 0.016 0.017 0.019 0.021 0.022 0.024 0.027 0.031 0.035
5 0.007 0.008 0.009 0.011 0.013 0.014 0.016 0.018 0.019 0.021 0.023 0.025 0.027 0.031 0.035 0.039
6 0.007 0.008 0.010 0.011 0.013 0.014 0.016 0.018 0.020 0.022 0.024 0.025 0.027 0.031 0.035 0.040
7 0.006 0.008 0.009 0.010 0.012 0.013 0.015 0.017 0.018 0.020 0.022 0.024 0.026 0.029 0.033 0.037
8 0.005 0.006 0.007 0.008 0.010 0.011 0.012 0.014 0.015 0.016 0.018 0.019 0.021 0.024 0.027 0.030
9 0.003 0.004 0.004 0.005 0.006 0.006 0.007 0.008 0.009 0.010 0.010 0.011 0.012 0.014 0.016 0.018
10 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
11 -0.012 -0.012 -0.013 -0.013 -0.014 -0.014 -0.014 -0.015 -0.015 -0.015 -0.016 -0.016 -0.016 -0.017 -0.017 -0.017
12 -0.027 -0.028 -0.029 -0.030 -0.030 -0.031 -0.031 -0.032 -0.033 -0.033 -0.033 -0.034 -0.034 -0.035 -0.036 -0.036
13 -0.044 -0.045 -0.046 -0.047 -0.048 -0.049 -0.049 -0.050 -0.050 -0.051 -0.052 -0.052 -0.053 -0.053 -0.054 -0.055
14 -0.061 -0.062 -0.062 -0.063 -0.064 -0.065 -0.065 -0.066 -0.067 -0.067 -0.068 -0.068 -0.069 -0.069 -0.070 -0.071
15 -0.073 -0.074 -0.075 -0.076 -0.076 -0.077 -0.078 -0.078 -0.079 -0.079 -0.080 -0.080 -0.080 -0.081 -0.082 -0.082
16 -0.080 -0.081 -0.082 -0.082 -0.083 -0.083 -0.084 -0.084 -0.084 -0.085 -0.085 -0.085 -0.086 -0.086 -0.087 -0.087
17 -0.079 -0.079 -0.080 -0.080 -0.081 -0.081 -0.081 -0.081 -0.082 -0.082 -0.082 -0.082 -0.083 -0.083 -0.083 -0.083
18 -0.067 -0.067 -0.067 -0.067 -0.068 -0.068 -0.068 -0.068 -0.068 -0.068 -0.068 -0.068 -0.069 -0.069 -0.069 -0.069
19 -0.041 -0.041 -0.041 -0.041 -0.042 -0.042 -0.042 -0.042 -0.042 -0.042 -0.042 -0.042 -0.042 -0.042 -0.042 -0.042
20 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
21 -0.041 -0.041 -0.041 -0.041 -0.042 -0.042 -0.042 -0.042 -0.042 -0.042 -0.042 -0.042 -0.042 -0.042 -0.042 -0.042
22 -0.067 -0.067 -0.067 -0.067 -0.068 -0.068 -0.068 -0.068 -0.068 -0.068 -0.068 -0.068 -0.069 -0.069 -0.069 -0.069
23 -0.079 -0.079 -0.080 -0.080 -0.081 -0.081 -0.081 -0.081 -0.082 -0.082 -0.082 -0.082 -0.083 -0.083 -0.083 -0.083
24 -0.080 -0.081 -0.082 -0.082 -0.083 -0.083 -0.084 -0.084 -0.084 -0.085 -0.085 -0.085 -0.086 -0.086 -0.087 -0.087
25 -0.073 -0.074 -0.075 -0.076 -0.076 -0.077 -0.078 -0.078 -0.079 -0.079 -0.080 -0.080 -0.080 -0.081 -0.082 -0.082
26 -0.061 -0.062 -0.062 -0.063 -0.064 -0.065 -0.065 -0.066 -0.067 -0.067 -0.068 -0.068 -0.069 -0.069 -0.070 -0.071
27 -0.044 -0.045 -0.046 -0.047 -0.048 -0.049 -0.049 -0.050 -0.050 -0.051 -0.052 -0.052 -0.053 -0.053 -0.054 -0.055
28 -0.027 -0.028 -0.029 -0.030 -0.030 -0.031 -0.031 -0.032 -0.033 -0.033 -0.033 -0.034 -0.034 -0.035 -0.036 -0.036
29 -0.012 -0.012 -0.013 -0.013 -0.014 -0.014 -0.014 -0.015 -0.015 -0.015 -0.016 -0.016 -0.016 -0.017 -0.017 -0.017
30 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
31 0.003 0.004 0.004 0.005 0.006 0.006 0.007 0.008 0.009 0.010 0.010 0.011 0.012 0.014 0.016 0.018
32 0.005 0.006 0.007 0.008 0.010 0.011 0.012 0.014 0.015 0.016 0.018 0.019 0.021 0.024 0.027 0.030
33 0.006 0.008 0.009 0.010 0.012 0.013 0.015 0.017 0.018 0.020 0.022 0.024 0.026 0.029 0.033 0.037
34 0.007 0.008 0.010 0.011 0.013 0.014 0.016 0.018 0.020 0.022 0.024 0.025 0.027 0.031 0.035 0.040
35 0.007 0.008 0.009 0.011 0.013 0.014 0.016 0.018 0.019 0.021 0.023 0.025 0.027 0.031 0.035 0.039
36 0.006 0.007 0.008 0.010 0.011 0.013 0.014 0.016 0.017 0.019 0.021 0.022 0.024 0.027 0.031 0.035
37 0.005 0.006 0.007 0.008 0.009 0.010 0.012 0.013 0.014 0.015 0.017 0.018 0.020 0.022 0.025 0.028
38 0.003 0.004 0.005 0.006 0.006 0.007 0.008 0.009 0.010 0.011 0.012 0.013 0.014 0.016 0.018 0.020
39 0.002 0.002 0.002 0.003 0.003 0.004 0.004 0.005 0.005 0.006 0.006 0.007 0.007 0.008 0.009 0.010
Posi
ção
da c
arga
Tabela 6 (Folha 2/3) - Viga contínua de quatro tramos com carga pontual
Relação dos vãos: n = L1 / LP L
Momento Flector MCFactor:
A B E
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 31 32 33 34 35 36 37 38 39 400
C
21 22 23 24 25 26 27 28 29 30
DL1 L L1L
P
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0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.10 1.20 1.30
0 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
1 -0.001 -0.001 -0.001 -0.001 -0.001 -0.001 -0.001 -0.001 -0.001 -0.002 -0.002 -0.002 -0.002 -0.002 -0.002 -0.002
2 -0.001 -0.001 -0.002 -0.002 -0.002 -0.002 -0.002 -0.003 -0.003 -0.003 -0.003 -0.003 -0.003 -0.004 -0.004 -0.004
3 -0.002 -0.002 -0.002 -0.003 -0.003 -0.003 -0.003 -0.004 -0.004 -0.004 -0.004 -0.005 -0.005 -0.005 -0.006 -0.006
4 -0.002 -0.002 -0.003 -0.003 -0.004 -0.004 -0.004 -0.005 -0.005 -0.005 -0.005 -0.006 -0.006 -0.007 -0.007 -0.008
5 -0.002 -0.003 -0.003 -0.004 -0.004 -0.004 -0.005 -0.005 -0.005 -0.006 -0.006 -0.006 -0.007 -0.007 -0.008 -0.008
6 -0.002 -0.003 -0.003 -0.004 -0.004 -0.004 -0.005 -0.005 -0.006 -0.006 -0.006 -0.007 -0.007 -0.007 -0.008 -0.009
7 -0.002 -0.003 -0.003 -0.003 -0.004 -0.004 -0.004 -0.005 -0.005 -0.005 -0.006 -0.006 -0.006 -0.007 -0.007 -0.008
8 -0.002 -0.002 -0.002 -0.003 -0.003 -0.003 -0.004 -0.004 -0.004 -0.004 -0.005 -0.005 -0.005 -0.006 -0.006 -0.006
9 -0.001 -0.001 -0.001 -0.002 -0.002 -0.002 -0.002 -0.002 -0.002 -0.003 -0.003 -0.003 -0.003 -0.003 -0.004 -0.004
10 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
11 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004
12 0.010 0.010 0.010 0.010 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.008 0.008 0.008
13 0.016 0.016 0.015 0.015 0.015 0.015 0.014 0.014 0.014 0.014 0.014 0.013 0.013 0.013 0.012 0.012
14 0.022 0.021 0.021 0.020 0.020 0.020 0.019 0.019 0.018 0.018 0.018 0.017 0.017 0.017 0.016 0.015
15 0.026 0.026 0.025 0.024 0.024 0.023 0.023 0.022 0.022 0.021 0.021 0.021 0.020 0.019 0.019 0.018
16 0.029 0.028 0.027 0.027 0.026 0.025 0.025 0.024 0.023 0.023 0.022 0.022 0.021 0.021 0.020 0.019
17 0.028 0.027 0.027 0.026 0.025 0.024 0.024 0.023 0.023 0.022 0.022 0.021 0.021 0.020 0.019 0.018
18 0.024 0.023 0.022 0.022 0.021 0.021 0.020 0.019 0.019 0.018 0.018 0.018 0.017 0.016 0.016 0.015
19 0.015 0.014 0.014 0.013 0.013 0.013 0.012 0.012 0.012 0.011 0.011 0.011 0.010 0.010 0.010 0.009
20 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
21 -0.021 -0.020 -0.019 -0.019 -0.018 -0.017 -0.017 -0.016 -0.016 -0.015 -0.015 -0.015 -0.014 -0.014 -0.013 -0.012
22 -0.045 -0.043 -0.042 -0.040 -0.039 -0.038 -0.037 -0.035 -0.034 -0.033 -0.033 -0.032 -0.031 -0.029 -0.028 -0.027
23 -0.069 -0.067 -0.064 -0.062 -0.060 -0.058 -0.056 -0.055 -0.053 -0.052 -0.050 -0.049 -0.048 -0.045 -0.043 -0.041
24 -0.091 -0.088 -0.085 -0.082 -0.079 -0.077 -0.074 -0.072 -0.070 -0.068 -0.066 -0.064 -0.063 -0.059 -0.057 -0.054
25 -0.108 -0.104 -0.100 -0.097 -0.093 -0.090 -0.087 -0.085 -0.082 -0.080 -0.078 -0.076 -0.074 -0.070 -0.067 -0.064
26 -0.116 -0.111 -0.107 -0.104 -0.100 -0.097 -0.094 -0.091 -0.088 -0.086 -0.083 -0.081 -0.079 -0.075 -0.071 -0.068
27 -0.112 -0.108 -0.104 -0.100 -0.097 -0.093 -0.091 -0.088 -0.085 -0.083 -0.080 -0.078 -0.076 -0.072 -0.069 -0.066
28 -0.093 -0.090 -0.086 -0.083 -0.081 -0.078 -0.075 -0.073 -0.071 -0.069 -0.067 -0.065 -0.063 -0.060 -0.057 -0.055
29 -0.057 -0.055 -0.053 -0.051 -0.049 -0.048 -0.046 -0.045 -0.043 -0.042 -0.041 -0.040 -0.039 -0.037 -0.035 -0.033
30 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
31 -0.011 -0.013 -0.016 -0.018 -0.021 -0.024 -0.027 -0.030 -0.033 -0.036 -0.039 -0.042 -0.046 -0.053 -0.060 -0.067
32 -0.018 -0.022 -0.026 -0.031 -0.035 -0.040 -0.045 -0.050 -0.055 -0.061 -0.066 -0.072 -0.077 -0.089 -0.100 -0.112
33 -0.023 -0.028 -0.033 -0.038 -0.044 -0.050 -0.056 -0.062 -0.069 -0.075 -0.082 -0.089 -0.096 -0.110 -0.124 -0.139
34 -0.024 -0.030 -0.035 -0.041 -0.047 -0.054 -0.060 -0.067 -0.074 -0.081 -0.088 -0.095 -0.103 -0.118 -0.134 -0.150
35 -0.024 -0.029 -0.034 -0.040 -0.046 -0.052 -0.059 -0.065 -0.072 -0.079 -0.086 -0.093 -0.100 -0.115 -0.131 -0.146
36 -0.021 -0.026 -0.031 -0.036 -0.041 -0.047 -0.053 -0.059 -0.065 -0.071 -0.077 -0.083 -0.090 -0.103 -0.117 -0.131
37 -0.017 -0.021 -0.025 -0.029 -0.034 -0.038 -0.043 -0.048 -0.052 -0.057 -0.063 -0.068 -0.073 -0.084 -0.095 -0.106
38 -0.012 -0.015 -0.018 -0.021 -0.024 -0.027 -0.030 -0.033 -0.037 -0.040 -0.044 -0.048 -0.051 -0.059 -0.067 -0.075
39 -0.006 -0.008 -0.009 -0.011 -0.012 -0.014 -0.016 -0.017 -0.019 -0.021 -0.023 -0.025 -0.027 -0.030 -0.034 -0.039
Posi
ção
da c
arga
Tabela 6 (Folha 3/3) - Viga contínua de quatro tramos com carga pontual
Relação dos vãos: n = L1 / LP L
Momento Flector MDFactor:
A B E
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 31 32 33 34 35 36 37 38 39 400
C
21 22 23 24 25 26 27 28 29 30
DL1 L L1L
P
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α α α α = 6 / 10 = 0.6
β β β β = 8 / 10 =0.8
p = 20 kN/m
s1 = s2 = s = 10 kN/m
Momentos Flectoresi)
MB = -0.068 p L 2 = -136 kNm
MC = -0.086 p L 2 = -172 kNm
M2 = (MB+MC)/2 + pL2/8 = +96 kNm
ii)
MB = -0.08 s L 2 = -80 kNm
MC = -0.047 s L 2 = -47 kNm
M2 = +62 kNm
i) + ii)
MB = -216 kNm
MC = -219 kNm
M2 = +158 kNm
Esforços Tranversosi)
VBdir = 2 * (M2 - MB) / 5 = +93 kN
VCesq = 2 * (MC - M2) / 5 = -107 kN
ii)
VBdir = 2 * (M2 - MB) / 5 = +57 kN
VCesq = 2 * (MC - M2) / 5 = -43 kN
i) + ii)
VBdir = +149 kN
VCesq = -151 kN
Exemplo (Tabela 7) - Viga contínua de três tramos com cargas distribuídas
+
=
=
+
MBMC
M2
VdirB
VesqC
MB MC
M2
VdirB
VesqC
MB MC
M2
VdirB
VesqC
A B D
s2
C
p
L = 61 L = 10 L = 82
10 11 12 13 14 15 16 17 18 19 20 22 23 24 25 26 27 28 29 3021
s1
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ββββp
s1
s2
s3
α βα βα βα β 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 α βα βα βα β 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
-0.070 -0.070 -0.069 -0.067 -0.064 -0.060 -0.055 -0.050 -0.043 -0.036 -0.070 -0.070 -0.074 -0.079 -0.087 -0.098 -0.111 -0.127 -0.145 -0.166
-0.007 -0.006 -0.006 -0.006 -0.006 -0.006 -0.006 -0.006 -0.006 -0.006 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.001 0.001 0.001
-0.066 -0.068 -0.069 -0.070 -0.072 -0.073 -0.074 -0.074 -0.075 -0.076 -0.066 -0.061 -0.057 -0.053 -0.050 -0.047 -0.044 -0.042 -0.040 -0.038
0.002 0.004 0.007 0.010 0.014 0.019 0.025 0.031 0.038 0.046 -0.007 -0.012 -0.019 -0.028 -0.039 -0.053 -0.069 -0.087 -0.107 -0.129
-0.070 -0.070 -0.069 -0.067 -0.065 -0.061 -0.057 -0.052 -0.046 -0.039 -0.070 -0.070 -0.073 -0.079 -0.087 -0.098 -0.111 -0.126 -0.145 -0.165
-0.012 -0.012 -0.012 -0.012 -0.011 -0.011 -0.011 -0.011 -0.011 -0.011 0.004 0.004 0.004 0.003 0.003 0.003 0.003 0.003 0.003 0.002
-0.061 -0.063 -0.064 -0.065 -0.066 -0.067 -0.068 -0.069 -0.070 -0.070 -0.068 -0.063 -0.058 -0.054 -0.051 -0.048 -0.045 -0.043 -0.041 -0.039
0.002 0.004 0.006 0.009 0.013 0.018 0.023 0.029 0.035 0.043 -0.006 -0.012 -0.019 -0.028 -0.039 -0.053 -0.068 -0.086 -0.106 -0.129
-0.074 -0.073 -0.072 -0.071 -0.068 -0.065 -0.061 -0.056 -0.050 -0.044 -0.069 -0.069 -0.072 -0.078 -0.086 -0.097 -0.110 -0.125 -0.144 -0.164
-0.019 -0.019 -0.019 -0.019 -0.018 -0.018 -0.018 -0.018 -0.018 -0.018 0.007 0.006 0.006 0.005 0.005 0.005 0.005 0.004 0.004 0.004
-0.057 -0.058 -0.060 -0.061 -0.062 -0.063 -0.064 -0.064 -0.065 -0.066 -0.069 -0.064 -0.060 -0.056 -0.052 -0.049 -0.047 -0.044 -0.042 -0.040
0.002 0.004 0.006 0.009 0.012 0.016 0.021 0.027 0.033 0.040 -0.006 -0.012 -0.019 -0.028 -0.039 -0.052 -0.068 -0.086 -0.106 -0.128
-0.079 -0.079 -0.078 -0.076 -0.074 -0.071 -0.067 -0.062 -0.057 -0.051 -0.067 -0.067 -0.071 -0.076 -0.084 -0.095 -0.108 -0.124 -0.142 -0.163
-0.028 -0.028 -0.028 -0.028 -0.027 -0.027 -0.027 -0.027 -0.027 -0.027 0.010 0.009 0.009 0.008 0.008 0.007 0.007 0.006 0.006 0.006
-0.053 -0.054 -0.056 -0.057 -0.058 -0.059 -0.060 -0.060 -0.061 -0.061 -0.070 -0.065 -0.061 -0.057 -0.053 -0.050 -0.048 -0.045 -0.043 -0.041
0.002 0.003 0.005 0.008 0.011 0.015 0.020 0.025 0.031 0.038 -0.006 -0.012 -0.019 -0.028 -0.039 -0.052 -0.067 -0.085 -0.105 -0.128
-0.087 -0.087 -0.086 -0.084 -0.082 -0.079 -0.076 -0.071 -0.066 -0.060 -0.064 -0.065 -0.068 -0.074 -0.082 -0.093 -0.106 -0.122 -0.140 -0.161
-0.039 -0.039 -0.039 -0.039 -0.039 -0.038 -0.038 -0.038 -0.038 -0.038 0.014 0.013 0.012 0.011 0.011 0.010 0.010 0.009 0.009 0.008
-0.050 -0.051 -0.052 -0.053 -0.054 -0.055 -0.056 -0.057 -0.057 -0.058 -0.072 -0.066 -0.062 -0.058 -0.054 -0.051 -0.049 -0.046 -0.044 -0.042
0.002 0.003 0.005 0.008 0.011 0.014 0.019 0.024 0.029 0.035 -0.006 -0.011 -0.018 -0.027 -0.039 -0.052 -0.067 -0.085 -0.105 -0.127
-0.098 -0.098 -0.097 -0.095 -0.093 -0.090 -0.087 -0.082 -0.078 -0.072 -0.060 -0.061 -0.065 -0.071 -0.079 -0.090 -0.103 -0.119 -0.137 -0.158
-0.053 -0.053 -0.052 -0.052 -0.052 -0.052 -0.051 -0.051 -0.051 -0.051 0.019 0.018 0.016 0.015 0.014 0.014 0.013 0.012 0.012 0.011
-0.047 -0.048 -0.049 -0.050 -0.051 -0.052 -0.053 -0.053 -0.054 -0.055 -0.073 -0.067 -0.063 -0.059 -0.055 -0.052 -0.049 -0.047 -0.045 -0.042
0.002 0.003 0.005 0.007 0.010 0.014 0.018 0.022 0.027 0.033 -0.006 -0.011 -0.018 -0.027 -0.038 -0.052 -0.067 -0.085 -0.104 -0.127
-0.111 -0.111 -0.110 -0.108 -0.106 -0.103 -0.100 -0.096 -0.091 -0.086 -0.055 -0.057 -0.061 -0.067 -0.076 -0.087 -0.100 -0.116 -0.134 -0.155
-0.069 -0.068 -0.068 -0.067 -0.067 -0.067 -0.067 -0.066 -0.066 -0.066 0.025 0.023 0.021 0.020 0.019 0.018 0.017 0.016 0.015 0.014
-0.044 -0.045 -0.047 -0.048 -0.049 -0.049 -0.050 -0.051 -0.051 -0.052 -0.074 -0.068 -0.064 -0.060 -0.056 -0.053 -0.050 -0.047 -0.045 -0.043
0.002 0.003 0.005 0.007 0.010 0.013 0.017 0.021 0.026 0.032 -0.006 -0.011 -0.018 -0.027 -0.038 -0.051 -0.067 -0.084 -0.104 -0.126
-0.127 -0.126 -0.125 -0.124 -0.122 -0.119 -0.116 -0.112 -0.108 -0.103 -0.050 -0.052 -0.056 -0.062 -0.071 -0.082 -0.096 -0.112 -0.131 -0.151
-0.087 -0.086 -0.086 -0.085 -0.085 -0.085 -0.084 -0.084 -0.084 -0.084 0.031 0.029 0.027 0.025 0.024 0.022 0.021 0.020 0.019 0.018
-0.042 -0.043 -0.044 -0.045 -0.046 -0.047 -0.047 -0.048 -0.049 -0.049 -0.074 -0.069 -0.064 -0.060 -0.057 -0.053 -0.051 -0.048 -0.046 -0.044
0.001 0.003 0.004 0.006 0.009 0.012 0.016 0.020 0.025 0.030 -0.006 -0.011 -0.018 -0.027 -0.038 -0.051 -0.066 -0.084 -0.104 -0.126
-0.145 -0.145 -0.144 -0.142 -0.140 -0.137 -0.134 -0.131 -0.126 -0.122 -0.043 -0.046 -0.050 -0.057 -0.066 -0.078 -0.091 -0.108 -0.126 -0.147
-0.107 -0.106 -0.106 -0.105 -0.105 -0.104 -0.104 -0.104 -0.104 -0.103 0.038 0.035 0.033 0.031 0.029 0.027 0.026 0.025 0.024 0.022
-0.040 -0.041 -0.042 -0.043 -0.044 -0.045 -0.045 -0.046 -0.046 -0.047 -0.075 -0.070 -0.065 -0.061 -0.057 -0.054 -0.051 -0.049 -0.046 -0.044
0.001 0.003 0.004 0.006 0.009 0.012 0.015 0.019 0.024 0.029 -0.006 -0.011 -0.018 -0.027 -0.038 -0.051 -0.066 -0.084 -0.104 -0.126
-0.166 -0.165 -0.164 -0.163 -0.161 -0.158 -0.155 -0.151 -0.147 -0.143 -0.036 -0.039 -0.044 -0.051 -0.060 -0.072 -0.086 -0.103 -0.122 -0.143
-0.129 -0.129 -0.128 -0.128 -0.127 -0.127 -0.126 -0.126 -0.126 -0.125 0.046 0.043 0.040 0.038 0.035 0.033 0.032 0.030 0.029 0.027
-0.038 -0.039 -0.040 -0.041 -0.042 -0.042 -0.043 -0.044 -0.044 -0.045 -0.076 -0.070 -0.066 -0.061 -0.058 -0.055 -0.052 -0.049 -0.047 -0.045
0.001 0.002 0.004 0.006 0.008 0.011 0.014 0.018 0.022 0.027 -0.006 -0.011 -0.018 -0.027 -0.038 -0.051 -0.066 -0.084 -0.103 -0.125
1.2
1.3
0.9
1
1.1
0.6
0.7
s1L2
pL2
0.4
0.5
0.4
0.5
1.3
0.8
0.9
1
1.1
1.2
Tabela 8 - Viga contínua de três tramos com cargas uniformemente distribuídas
MB MC
0.8
0.6
0.7
αααα
Factor multiplicativo
s3L2
s2L2
A B DαL L
s1s2
s3
C
p
βL
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 300
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n = 8 / 10 = 0.8
p = 20 kN/m
i) Identificação dos graus de
liberdade independentes
ii) Matrix de Rigidez
KBA = 3 EI / L1 = 3 EI / 8 = 0.375 EI
KBC = 3 EI / L2 = 3 EI / 10 = 0.300 EI
K11 = KBC + KBC = 0.675 EI
iii) Esforços de Fixação
M0BA = -20 x 82 / 8 = -160 kNm
M0BC = -20 x 102 / 8 = -250 kNm
Esforços desequilibrados
Q01 = - 160 + 250 = 90 kNm
iv) Solução da equação do Método
dos Deslocamentos
K11 x θθθθ1 + Q01 = 0
0.675 EI x θθθθ1 + 90 = 0
θθθθ1 = -133.3 / EI
v) Variação de momentos devidos
aos desloc. independentes
∆∆∆∆MBA = KBA θθθθ1 = 0.375EI (-133.3/EI)
∆∆∆∆MBA = - 50 kNm
∆∆∆∆MBC = KBC θθθθ1 = 0.300EI (-133.3/EI)
∆∆∆∆MBA = 40 kNm
vi) = ii) + v) Esforços Finais
MB = M0BA + ∆∆∆∆MBA = M0
BC + MBC
MB = -210 kNm
Exemplo de aplicação do método dos deslocamentos a uma viga contínua de dois tramos
(Considera-se a mesma estrutura do exemplo da Tabela 1 e o carregamento uniforme em toda a viga)
M0BA
KBC
KBA
M0BC
Q01
A B CL = 81 L = 10
p
10 11 12 13 14 15 16 17 18 19 201 2 3 4 5 6 7 8 90
θ1 = θB
∆ MBA
θ1 = 1
θ1 = -133.3 / EI
∆ MBC
MB
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n = 8 / 10 = 0.8
p = 20 kN/m
i) Matrix de Rigidez
KBA = 3 EI / L1 = 3 EI / 8 = 0.375 EI
KBC = 3 EI / L2 = 3 EI / 10 = 0.300 EI
K11 = KBC + KBC = 0.675 EI
ii) Coeficientes de repartição
C1 = KBA / K11 = 0.375/0.675 = 0.550
C2 = KBC / K11 = 0.300/0.675 = 0.444
iii) Momentos de fixação
M0BA = -20 x 82 / 8 = -160 kNm
M0BC = -20 x 102 / 8 = -250 kNm
iv) Repartição do momento
desequilibrado
∆∆∆∆M1 = -160 + 250 = 90
∆∆∆∆MBA = 0.556 x (-90) = -50
∆∆∆∆MBC = 0.444 x (-90) = -40
MB = -210 kNm
Nota: tratando-se de uma estrutura com 1 grau de liberdade, a aplicação do método de cross não é mais do que a repartição momento desequilibrado
proporcionalmente à rigidez das barras. Uma vez que a rigidez de rotação do nó não é mais do que a soma da rigidez das barras adjacentes (rigidez em
série).
Exemplo de aplicação do método de Cross a uma viga contínua de dois tramos
(Considera-se a mesma estrutura do exemplo da Tabela 1 e o carregamento uniforme em toda a viga)
C1 C2
A B CL = 81 L = 10
p
10 11 12 13 14 15 16 17 18 19 201 2 3 4 5 6 7 8 90
θ1 = θB
M0BA
M0BC
0.556 0.444
-160 250-50 -40
-210 -210210
MB
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n = 8 / 10 = 0.8
p = 20 kN/m
i) Momentos totais
Mf = M0 + MH
ii)
ViAB = - Vi
BA = 4 x 20 = 80 kN
ViBC = - Vi
CB = 5 x 20 = 100 kN
iii)
VHAB = - 210/8 = -26.25 kN
VHBC = - 210/10 = -21.00 kN
iv) Esforços transversos totais
Vf = V0 + VH
VAB = 53.75 kN
VBA = -106.25 kN
VBC = 121.0 kN
VCB = 79.0 kN
v)
RA = VAB = 53.75 kN
RB = - VBA + VBC = 227.25 kN
RC = - VCB = 79.0 kN
iv) Momentos positivos no vão
x1 = 53.75 / 20 = 2.69 m
M+BA = 20x(2x 2.69)2/8 = 72.4 kNm
x2 = 79.0 / 20 = 3.95 m
M+BC = 20x(2x 3.95)2/8 = 156.0 kNm
As reacções podem ser obtidas directamente do esforço transverso
(Considera-se a mesma estrutura do exemplo da Tabela 1 e o carregamento uniforme em toda a viga)
+
=
Exemplo de cálculo dos momentos positivos e dos esforços transversos numa viga de dois tramos
Momentos e esforços transversos isostáticos de cada tramo
Momento e esforço transverso hiperestático (esforço devido ao momento sobre o apoio)
A B CL = 81 L = 10
p
10 11 12 13 14 15 16 17 18 19 201 2 3 4 5 6 7 8 90
M+BC
ViAB
ViBC
ViBC
ViBA
VHAB
VHBC
-210
MB
VA
VesqB
VdirB
M+AB
M+BC
x1 x2
M+AB
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n = 8 / 10 = 0.8
p = 20 kN/m
i) Diagrama de esforços
MB = -210 kNm
M+máx = MD = 156 kNm
ii) Deformada
iii) Aplicação do PTV
MD = 1 x 6.05 x 3.95 / 10 = 2.39 m
ii) δδδδ = 1300.7 / EI
= 1 / EI x (-(10+3.95) / 6 x 210 x 2.39 + 10 / 3 x 250 x 2.39 x (1 + 6.05 x 3.05 / 102))
= 1 / EI x ( -1166.9 + 2467.6) = 1300.7 / EI
Exemplo de cálculo de deslocamentos no vão de um tramo de uma viga continua de dois tramos
(Considera-se a mesma estrutura do exemplo da Tabela 1 e o carregamento uniforme em toda a viga)
Cálculo do deslocamento no ponto D
A B CL = 81 L = 10
p
10 11 12 13 14 15 16 17 18 19 201 2 3 4 5 6 7 8 90
210
156 2.39
210
2.39 2.39250
+=EI 1
EI 1
dxEI
MML
∫=0
δ
16.05 3.95
D
MB
MD
δ
MD
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