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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```