Momentos de Engastamento Perfeito
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Momentos de Engastamento Perfeito


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UNIVERSIDADE DE SÃO PAULO 
ESCOLA DE ENGENHARIA DE SÃO CARLOS 
Departamento de Engenharia de Estruturas 
 
 
 
 
 
 
 
 
TABELAS DE VIGAS: 
Deslocamentos e Momentos de Engastamento Perfeito 
 
 
 
 
 
 
Revisão e adaptação: 
 
Libânio Miranda Pinheiro 
Bruna Catoia 
Thiago Catoia 
 
 
Colaboração: 
 
Marcos Vinicius Natal Moreira 
 
 
 
 
 
 
 
 
São Carlos, fevereiro de 2010 
 
 
TABELA 3.1a 
DESLOCAMENTOS ELÁSTICOS EM VIGAS 
FLECHA 
CASO 
VINCULAÇÃO E 
CARREGAMENTO wmax x 
EQUAÇÃO DA ELÁSTICA 
1 
p
\uf06cx EI
p
8
1 4\uf06c
 0 \uf028 \uf02934
EI24
p 4
4
\uf02b\uf061\uf02d\uf061
\uf06c
 
2 
p
\uf06c EI
p
30
1 4\uf06c
 0 \uf028 \uf02945
EI120
p 5
4
\uf02b\uf061\uf02d\uf061
\uf06c
 
3 
p
\uf06c EI
p
120
11 4\uf06c
 0 \uf028 \uf02911155
EI120
p 45
4
\uf02b\uf061\uf02d\uf061\uf02b\uf061\uf02d
\uf06c
 
4 
\uf06c
P
 EI
P
3
1 3\uf06c
 0 \uf028 \uf02923
EI6
P 3
3
\uf02b\uf061\uf02d\uf061
\uf06c
 
5 
\uf06c
M
 EI
M
2
1 2\uf06c
 0 \uf028 \uf0292
2
1
EI2
M
\uf061\uf02d
\uf06c
 
6 
\uf06c
p
x 
EI
p
384
5 4\uf06c
 \uf06c5,0 \uf028 \uf02912
EI24
p 23
4
\uf02b\uf061\uf02d\uf061
\uf061\uf06c
 
7 
p
\uf06c 
(*)4
EI
p
460
3 \uf06c
 \uf06c519,0 \uf028 \uf0297103
EI360
p 24
4
\uf02b\uf061\uf02d\uf061
\uf061\uf06c
 
8 
\uf06c
p
 
EI
p
120
1 4\uf06c
 \uf06c5,0 \uf028 \uf029
(**)
24
4
254016
EI960
p
\uf02b\uf061\uf02d\uf061
\uf061\uf06c
 
9 
P
\uf06c
2
\uf06c
2 
EI
P
48
1 3\uf06c
 \uf06c5,0 \uf028 \uf029
(**)
2
3
34
EI48
P
\uf02b\uf061\uf02d
\uf061\uf06c
 
10 
P
a b
x 
)ba( \uf0b3 
322
3
b
EI3
Pb
\uf0f7\uf0f7
\uf0f8
\uf0f6
\uf0e7\uf0e7
\uf0e8
\uf0e6 \uf02d\uf06c
\uf06c
 \uf0f7
\uf0f7
\uf0f8
\uf0f6
\uf0e7\uf0e7
\uf0e8
\uf0e6 \uf02d
3
b22\uf06c
\uf028 \uf029222 xb
EI6
Pbx
:ax \uf02d\uf02d\uf03c \uf06c
\uf06c
 
\uf06cEI3
bPa
:ax
22
\uf03d 
\uf028 \uf02922 xax2
EI6
)x(Pa
:ax \uf02d\uf02d
\uf02d
\uf03e \uf06c
\uf06c
\uf06c
 
11 
\uf06c
M
 
EI
M
39
1 2\uf06c
 \uf06c423,0 \uf028 \uf02923
EI6
M 2
2
\uf02b\uf061\uf02d\uf061
\uf061\uf06c
 
12 
M
a b
 
)423,0a( \uf06c\uf0b3 
3
2
2
b
3EI3
M
\uf0f7\uf0f7
\uf0f8
\uf0f6
\uf0e7\uf0e7
\uf0e8
\uf0e6
\uf02d
\uf06c
\uf06c
 
\uf0f7\uf0f7
\uf0f8
\uf0f6
\uf0e7\uf0e7
\uf0e8
\uf0e6
\uf02d 2
2
b
3
\uf06c
 
\uf028 \uf029222 xb3
EI6
Mx
:ax \uf02d\uf02d\uf03c \uf06c
\uf06c
 
\uf028 \uf029x2a3x
EI6
)x(M
:ax 22 \uf06c
\uf06c
\uf02d\uf02b
\uf02d
\uf03e 
Extraída de ISNARD; GREKOW; MROZOWICZ (1971) e de SCHIEL (1976). 
Revista e adaptada por Libânio M. Pinheiro, Bruna Catoia e Thiago Catoia. 
\uf06c/x\uf03d\uf061 (*) Valor aproximado (**) 5,0\uf0a3\uf061 
 
 
TABELA 3.1b 
DESLOCAMENTOS ELÁSTICOS EM VIGAS 
FLECHA 
CASO 
VINCULAÇÃO E 
CARREGAMENTO wmax x 
EQUAÇÃO DA ELÁSTICA 
13 
\uf06c
M M
x 
EI8
M 2\uf06c
 \uf06c5,0 \uf028 \uf029\uf061\uf02d\uf061 1
EI2
M 2\uf06c
 
14 
\uf06c
p
x 
(*)4
EI
p
554
3 \uf06c
 
 
\uf06c422,0
 
\uf028 \uf029\uf061\uf02b\uf061\uf02d\uf061 34
4
32
EI48
p\uf06c
 
15 
p
\uf06c 
(*)4
EI
p
1258
3 \uf06c
 
 
\uf06c447,0
 
\uf028 \uf029\uf061\uf02b\uf061\uf02d\uf061 35
4
2
EI120
p\uf06c
 
16 
p
\uf06c 
(*)4
EI
p
328
1 \uf06c
 
 
\uf06c402,0
 
\uf028 \uf029\uf061\uf02b\uf061\uf02d\uf061\uf02b\uf061\uf02d 311102
EI240
p 345
4\uf06c
 
17 
\uf06c
M
 EI27
M 2\uf06c
 \uf06c
3
1
 \uf028 \uf029\uf061\uf02b\uf061\uf02d\uf061 23
2
2
EI4
M\uf06c
 
18 
\uf06c
p
x 
EI
p
384
1 4\uf06c
 \uf06c5,0 \uf028 \uf029234
4
2
EI24
p
\uf061\uf02b\uf061\uf02d\uf061
\uf06c
 
19 
p
\uf06c 
(*)4
EI
p
764
1 \uf06c
 
 
\uf06c525,0
 
\uf028 \uf029235
4
23
EI120
p
\uf061\uf02b\uf061\uf02d\uf061
\uf06c
 
20 
\uf06c
p
 
EI
p
3840
7 4\uf06c
 \uf06c5,0 \uf028 \uf029
(**)
235
4
254016
EI960
p
\uf061\uf02b\uf061\uf02d\uf061
\uf06c
 
21 
P
\uf06c
2 2
\uf06c
 
EI
P
192
1 3\uf06c
 \uf06c5,0 \uf028 \uf029
(**)
23
3
34
EI48
P
\uf061\uf02b\uf061\uf02d
\uf06c
 
22 
p
a a\uf06c
x 
\uf028 \uf029332 a3a6
EI24
pa
\uf06c\uf06c \uf02d\uf02b
\uf028 \uf02922
2
a245
EI384
p
\uf02d\uf06c
\uf06c
 
a\uf02d 
 
 
\uf06c5,0 
\uf028 \uf029\uf06c\uf06c 23223 a6xa6ax4x
EI24
px
:0x
\uf02d\uf02b\uf02b\uf02b
\uf03c
\uf028 \uf02932223 a6xa6x2x
EI24
px
:x0
\uf06c\uf06c\uf06c
\uf06c
\uf02b\uf02d\uf02b\uf02d
\uf03c\uf03c
23 
 
\uf06ca a
PP
x 
\uf028 \uf029\uf06c3a2
EI6
Pa 2
\uf02b 
 
EI8
Pa 2\uf06c
\uf02d 
a\uf02d 
 
 
\uf06c5,0 
\uf028 \uf029\uf06ca3ax3x
EI6
Px
:0x 2 \uf02d\uf02b\uf03c 
 
\uf028 \uf029\uf06c\uf06c \uf02d\uf03c\uf03c xx
EI2
Pa
:x0 
Extraída de ISNARD; GREKOW; MROZOWICZ (1971) e de SCHIEL (1976). 
Revista e adaptada por Libânio M. Pinheiro, Bruna Catoia e Thiago Catoia. 
\uf06c/x\uf03d\uf061 (*) Valor aproximado (**) 5,0\uf0a3\uf061 
 
 
TABELA 3.1c 
DESLOCAMENTOS ELÁSTICOS EM VIGAS 
FLECHA 
CASO 
VINCULAÇÃO E 
CARREGAMENTO wmax x 
24 
p
x
a b
 
\uf028 \uf029434 bb43
EI24
p
\uf02b\uf02d \uf06c\uf06c 0 
25 
p
a b 
\uf028 \uf029323 aa1020
EI120
pa
\uf02b\uf02d \uf06c\uf06c 0 
26 
p
a b
c/2 c/2
 
(*)32
2
64
c
4
c
a2a2
ab
EI6
pc
\uf02b
\uf0fa
\uf0fa
\uf0fb
\uf0f9
\uf0ea
\uf0ea
\uf0eb
\uf0e9
\uf0f7\uf0f7
\uf0f8
\uf0f6
\uf0e7\uf0e7
\uf0e8
\uf0e6
\uf02d\uf02d\uf06c
\uf06c
 a 
27 
P
a b
P
a
x 
\uf028 \uf02922 a43
EI24
Pa
\uf02d\uf06c \uf06c5,0 
28 
P P
\uf06c
3
\uf06c
3
\uf06c
3 
EI
P
648
23 3\uf06c
 \uf06c5,0 
29 
P P
4
\uf06c
4 4
P
4
\uf06c \uf06c \uf06c
 
EI
P
384
19 3\uf06c
 \uf06c5,0 
30 
P P
55 5
P
5
P
5
\uf06c \uf06c \uf06c \uf06c \uf06c
 
EI
P
1000
63 3\uf06c
 \uf06c5,0 
31 
P
22
\uf06c \uf06c
 EI
P
240
5 3\uf06c
 \uf06c447,0 
32 
P P
ba a
x 
EI
bPa
24
1 2
 \uf06c5,0 
33 
p
a \uf06c
x 
\uf028 \uf029323 a4a3
EI24
pa
\uf06c\uf06c \uf02d\uf02b a\uf02d 
34 
p
a \uf06c
x \uf028 \uf029
\uf028 \uf02912a7a15a20
aEI360
p 3224 \uf02b\uf02d\uf02d
\uf02b
\uf06c\uf06c
\uf06c
\uf06c
 a\uf02d 
35 
P
a \uf06c
x 
\uf028 \uf029\uf06c\uf02ba
EI3
Pa 2
 a\uf02d 
36 
p
a \uf06c 
\uf028 \uf029323 a6a6
EI48
pa
\uf06c\uf06c \uf02d\uf02b a\uf02d 
37 
P
a \uf06c
 
\uf028 \uf0293a4
EI6
Pa 2
\uf02b a\uf02d 
38 
a
M
\uf06c
x 
\uf028 \uf029a2
EI4
Ma
\uf02b\uf06c a\uf02d 
Extraída de ISNARD; GREKOW; MROZOWICZ (1971). 
Revista e adaptada por Libânio M. Pinheiro, Bruna Catoia e Thiago Catoia. 
(*) Não corresponde necessariamente ao deslocamento máximo 
 
 
TABELA 3.2a 
MOMENTOS DE ENGASTAMENTO PERFEITO 
A B\uf06c C D
\uf06c
 FE
\uf06cCARREGAMENTO 
MBA MCD MDC MEF 
1 
p
 8
p 2\uf06c
\uf02d 
12
p 2\uf06c
 
12
p 2\uf06c
\uf02d 
8
p 2\uf06c
 
2 
p
a c a
 
\uf028 \uf02922 c3
16
pc
\uf02d\uf02d \uf06c
\uf06c
 \uf028 \uf02922 c3
24
pc
\uf02d\uf06c
\uf06c
 \uf028 \uf02922 c3
24
pc
\uf02d\uf02d \uf06c
\uf06c
 \uf028 \uf02922 c3
16
pc
\uf02d\uf06c
\uf06c
 
3 
p
c b
 
\uf028 \uf02922
2
2
c2
8
pc
\uf02d\uf02d \uf06c
\uf06c
\uf028 \uf02922
2
2
cbc4b6
12
pc
\uf02b\uf02b
\uf06c
 \uf028 \uf0292
2
2
cbc4
12
pc
\uf02b\uf02d
\uf06c
 \uf028 \uf0292
2
2
b
8
pc
\uf02b\uf06c
\uf06c
 
4 
\uf06c/2
p
\uf06c/2
 
2p
128
7
\uf06c\uf02d 2p
192
11
\uf06c 2p
192
5
\uf06c\uf02d 2p
128
9
\uf06c 
5 
p
ca 
\uf028 \uf0292
2
2
a
8
pc
\uf02b\uf02d \uf06c
\uf06c
 \uf028 \uf0292
2
2
cac4
12
pc
\uf02b
\uf06c
 \uf028 \uf029222
2
cac4a6
12
pc
\uf02b\uf02b\uf02d
\uf06c
 \uf028 \uf02922
2
2
c2
8
pc
\uf02d\uf06c
\uf06c
 
6 
p
\uf06c/2 \uf06c/2 
2p
128
9
\uf06c\uf02d 2p
192
5
\uf06c 2p
192
11
\uf06c\uf02d 2p
128
7
\uf06c 
7 aa
p
 
\uf028 \uf029a23
4
pa 2
\uf02d\uf02d \uf06c
\uf06c
 \uf028 \uf029a23
6
pa 2
\uf02d\uf06c
\uf06c
 \uf028 \uf029a23
6
pa 2
\uf02d\uf02d \uf06c
\uf06c
 \uf028 \uf029a23
4
pa 2
\uf02d\uf06c
\uf06c
 
8 
aa
p
p
 
\uf028 \uf029a23
4
pa 2
\uf02d\uf02d \uf06c
\uf06c
 \uf028 \uf0292
2
2
a
2
pa
\uf02d\uf06c
\uf06c
 \uf028 \uf0292
2
2
a
2
pa
\uf02d\uf02d \uf06c
\uf06c
 \uf028 \uf029a23
4
pa 2
\uf02d\uf06c
\uf06c
 
9 
a
P
b 
\uf028 \uf029a
2
Pab
2
\uf02b\uf02d \uf06c
\uf06c
 
2
2Pab
\uf06c
 
2
2bPa
\uf06c
\uf02d \uf028 \uf029b
2
Pab
2
\uf02b\uf06c
\uf06c
 
10 
P
\uf06c/2 \uf06c/2 
16
P3 \uf06c
\uf02d 
8
P\uf06c
 
8
P\uf06c
\uf02d 
16
P3 \uf06c
 
11 
aa
P P
 
\uf028 \uf029a
2
Pa3
\uf02d\uf02d \uf06c
\uf06c
 \uf028 \uf029aPa \uf02d\uf06c
\uf06c
 \uf028 \uf029aPa \uf02d\uf02d \uf06c
\uf06c
 \uf028 \uf029a
2
Pa3
\uf02d\uf06c
\uf06c
 
12 
P P
\uf06c/3 \uf06c/3 \uf06c/3 3
P\uf06c
\uf02d 
9
P2 \uf06c
 
9
P2 \uf06c
\uf02d 
3
P\uf06c
 
13 
P P
\uf06c/4
P
\uf06c/4 \uf06c/4 \uf06c/4 32
P15 \uf06c
\uf02d 
16
P5 \uf06c
 
16
P5 \uf06c
\uf02d 
32
P15 \uf06c
 
14 
P P
a
P P (*)
a a a a
 
\uf028 \uf0291n
n8
P 2 \uf02d\uf02d
\uf06c
 \uf028 \uf0291n
n12
P 2 \uf02d
\uf06c
 \uf028 \uf0291n
n12
P 2 \uf02d\uf02d
\uf06c
 \uf028 \uf0291n
n8
P 2 \uf02d
\uf06c
 
15 
a b
M
 
\uf028 \uf02922
2
a3
2
M
\uf02d\uf02d \uf06c
\uf06c
\uf028 \uf029\uf06c
\uf06c
2b3
Mb
2
\uf02d \uf028 \uf029a32Ma
2
\uf02d\uf02d \uf06c
\uf06c
 \uf028 \uf02922
2
b3
2
M
\uf06c
\uf06c
\uf02d 
16 
P P
a/2
P P (*)
a a a a/2
 
\uf028 \uf0291n2
n16
P 2 \uf02b\uf02d
\uf06c
 \uf028 \uf0291n2
n24
P 2 \uf02b
\uf06c
 \uf028 \uf0291n2
n24
P 2 \uf02b\uf02d
\uf06c
 \uf028 \uf0291n2
n16
P 2 \uf02b
\uf06c 
Extraída de SOUZA; ANTUNES (1983), JIMENES MONTOYA; GARCIA MESEGUER; MORAN CABRE (1973) e 
de SCHREYER (1965). Convenção de GRINTER. (*) a/n \uf06c\uf03d 
Revista e adaptada por Libânio M. Pinheiro, Bruna Catoia e Thiago Catoia. 
TABELA 3.2b 
MOMENTOS DE ENGASTAMENTO PERFEITO 
A B\uf06c C D
\uf06c
 FE
\uf06c
 CARREGAMENTO 
MBA MCD MDC MEF 
17 
p
 15
p 2\uf06c
\uf02d 
30
p 2\uf06c
 
20
p 2\uf06c
\uf02d 
120
p7 2\uf06c
 
18 
p
 120
p7 2\uf06c
\uf02d 
20
p 2\uf06c
 
30
p 2\uf06c
\uf02d 
15
p 2\uf06c
 
19 
p
\uf06c/2 \uf06c/2 
2p
480
17
\uf06c\uf02d 
30
p 2\uf06c
 
2p
160
3
\uf06c\uf02d 2p
960
41
\uf06c 
20 
p
\uf06c/2 \uf06c/2
 
2p
960
41
\uf06c\uf02d 2p
160
3
\uf06c 
30
p 2\uf06c
\uf02d 2p
480
17
\uf06c 
21 
p
\uf06c/2 \uf06c/2 
2p
1920
53
\uf06c\uf02d 2p
960
7
\uf06c 2p
960
23
\uf06c\uf02d 2p
1920
37
\uf06c 
22 
p
\uf06c/2 \uf06c/2 
2p
1920
37
\uf06c\uf02d 2p
960
23
\uf06c 2p
960
7
\uf06c\uf02d 2p
1920
53
\uf06c 
23 
p
\uf06c/2 \uf06c/2
 
2p
64
5
\uf06c\uf02d 2p
96
5
\uf06c 2p
96
5
\uf06c\uf02d 2p
64
5
\uf06c 
24 
p p
\uf06c/2 \uf06c/2
 
2p
64
3
\uf06c\uf02d 
32
p 2\uf06c
 
32
p 2\uf06c
\uf02d 2p
64
3
\uf06c 
25 parábola p
 
2p
120
11
\uf06c\uf02d 
20
p 2\uf06c
 
15
p 2\uf06c
\uf02d 
12
p 2\uf06c
 
26 
parábola p
 24
p 2\uf06c
\uf02d 
60
p 2\uf06c
 
30
p 2\uf06c
\uf02d 
30
p 2\uf06c
 
27 
pparábola
 10
p 2\uf06c
\uf02d 
15
p 2\uf06c
 
15
p 2\uf06c
\uf02d 
10
p 2\uf06c
 
28 
p
parábola
p
\uf06c/2 \uf06c/2 
2p
80
7
\uf06c\uf02d 2p
120
7
\uf06c 2p
120
7
\uf06c\uf02d 2p
80
7
\uf06c 
29 a EI
a3
2\uf06c
\uf02b EI
a6
2\uf06c
\uf02b EI
a6
2\uf06c
\uf02b EI
a3
2\uf06c
\uf02b 
30 
\uf061
 
EI
3
\uf06c
\uf061
\uf02b EI
2
\uf06c
\uf061
\uf02b EI
4
\uf06c
\uf061
\uf02b --- 
31 
\uf061
 
--- EI
4
\uf06c
\uf061
\uf02d EI
2
\uf06c
\uf061
\uf02d EI
3
\uf06c
\uf061
\uf02d 
32 
h
t+\uf044t
t 
t
h2
EI3
t\uf044\uf061\uf02b th
EI
t\uf044\uf061\uf02d th
EI
t\uf044\uf061\uf02b th2
EI3
t\uf044\uf061\uf02d 
Extraída de SOUZA; ANTUNES (1983), JIMENES MONTOYA; GARCIA MESEGUER; MORAN CABRE (1973) 
e de SCHREYER (1965). Convenção de GRINTER. 
Revista e adaptada por Libânio M. Pinheiro, Bruna Catoia e Thiago Catoia. 
 
TABELA 3.2c 
MOMENTOS DE ENGASTAMENTO PERFEITO 
CARREGAMENTO 
A B\uf06c C D
\uf06c
 FE
\uf06c
 
\uf05b \uf05d)ca2(c2)ca(a
8
p
M 244
2BA
\uf02b\uf02b\uf02b\uf02d\uf02d\uf03d \uf06c
\uf06c
 
\uf05b \uf05d \uf05b \uf05d\uf07b \uf07d4433
2CD
b)cb(3b)cb(4
12
p
M \uf02d\uf02b\uf02d\uf02d\uf02b\uf03d \uf06c
\uf06c
 
\uf05b \uf05d \uf05b \uf05d\uf07b \uf07d4433
2DC
a)ca(3a)ca(4
12
p
M \uf02d\uf02b\uf02d\uf02d\uf02b\uf02d\uf03d \uf06c
\uf06c
 
33 
p
a c b
 
\uf05b \uf05d)cb2(c2)cb(b
8
p