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resmat2007a


DisciplinaResistência dos Materiais II4.519 materiais116.590 seguidores
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3.105: Variac¸a\u2dco do Momento Esta´tico
3.4.2 Tenso\u2dces de Cisalhamento em Vigas de Sec¸a\u2dco Retangular
Constante
Sejam conhecidos o DMF e o DEC da viga. Na figura 3.106 representamos uma viga
bi-apoiada, mas o sistema de apoios poderia ser qualquer.
O elemento de volume de comprimento elementar dx, limitado pelas sec¸o\u2dces de abscissas
x e x+ dx e o elemento de a´rea dy × dz em torno de um ponto P(y, z) gene´rico da sec¸a\u2dco
determinam um elemento de volume dx× dy × dz.
6
6
6
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd
x dx
dy
dz
dx
dA
y
z
z
y
dA
P
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
6
Figura 3.106: Viga bi-apoiada
Nas faces direita e esquerda, \u3c4xy = \u3c4 e´ a tensa\u2dco tangencial na sec¸a\u2dco transversal.
Nas faces superior e inferior, \u3c4yx = \u3c4 e´ a tensa\u2dco tangencial nos planos longitudinais.
A existe\u2c6ncia de tenso\u2dces de cisalhamento em planos longitudinais e´ verificada em vigas
constituidas de elementos longitudinais, conforme a figura 3.107.
Para o ca´lculo das tenso\u2dces de cisalhamento, ale´m das hipo´teses admitidas na ana´lise
das tenso\u2dces normais de flexa\u2dco, admitimos a seguinte hipo´tese ba´sica
101
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
Figura 3.107: Viga constitu´\u131da de elementos longitudinais
\u2022 A tensa\u2dco de cisalhamento \u3c4 e´ constante na largura da sec¸a\u2dco.
Portanto \u3c4 = \u3c4(y) somente, isto e´, \u3c4 na\u2dco depende de z.
Seja uma camada de fibras AB//LN, de ordenada y, isto e´,uma camada de fibras
longitudinais // a` superf´\u131cie neutra conforme destaca figura 3.108.
\ufffd\ufffd
\ufffd\ufffd
\ufffd\ufffd
\ufffd\ufffd
\ufffd\ufffd
\ufffd\ufffd
A
LN y A
B
\u3c4
Figura 3.108: Tensa\u2dco tangencial constante na largura da viga
Nas figuras 3.109 e 3.110 destacamos a porc¸a\u2dco da viga, superior a esta camada, para
mostrar a tensa\u2dco tangencial (transversal e longitudinal) em uma sec¸a\u2dco S, sendo \u3c4 constante
de A ate´ B.
A resultante na direc¸a\u2dco longitudinal nas duas faces da figura 3.109 fornece:
F =
\u222b
Ai
\u3c3xdA\u21d2 e´ a resultante das tenso\u2dces normais na face esquerda.
F + dF =
\u222b
Ai
(\u3c3x + d\u3c3x)dA\u21d2 e´ a resultante das tenso\u2dces normais na face direita.
(3.97)
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd
M M +dM
Q Q +dQ \u3c3x \u3c3x + dx
Figura 3.109: Tenso\u2dces normais na flexa\u2dco
A condic¸a\u2dco de equil´\u131brio e´ a existe\u2c6ncia da forc¸a dF no plano longitudinal superior, de
a´rea bdx. Portanto:
dF = \u3c4xybdx =
\u222b
Ai
d\u3c3xdA =
\u222b
Ai
dM
I
ydA (3.98)
102
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
x
dF
\u3c3
dx
x d+ x\u3c3
F+dFF
dF
dx
Figura 3.110: Equil´\u131brio de forc¸as
obte´m -se:
\u3c4xy = \u3c4 =
1
Izb
dM
dx
\u222b
Ai
ydA\ufe38 \ufe37\ufe37 \ufe38
Ms
(3.99)
lembrando que dM
dx
= Q (esforc¸o cortante Q = Qy) tem-se enta\u2dco:
\u3c4 = \u3c4xy =
QMs
Izb
(3.100)
Do exerc´\u131cio preliminar: Ms = f(y) =
b
2
[
(h
2
)2 \u2212 y2
]
para´bola de 20, enta\u2dco a variac¸a\u2dco
de \u3c4 = \u3c4(y) e´ tambe´m uma para´bola do 20 grau.
Numa sec¸a\u2dco retangular enta\u2dco tem-se
y = 0\u21d2Mmaxs =
bh2
8
\u21d2 \u3c4max = Qbh
2/8
bbh3/12
=
3
2
Q
bh
(3.101)
Isto e´:
\u3c4max = 1, 5
Q
A
onde A = bh e´ a a´rea da sec¸a\u2dco.
Observe que \u3c4max = 1, 5\u3c4med (50% maior que \u3c4med =
Q
A
)
\u3c4max
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd
Superficie
de tensoes
parabolica
~ 
 \u2019
Solido de tensoes~ \u2019
\u3c4
solido de tensoes
Diagrama de tensoes
Vista de perfil do 
\u2019
~ 
~ 
Figura 3.111: So´lido de tenso\u2dces
Exerc´\u131cio Verificar a propriedade: Q =
\u222b
A \u3c4dA, que na\u2dco foi usada para calcular a tensa\u2dco
de cisalhamento \u3c4 .
Fac¸a
\u3c4 =
Q
Izb
b
2
\uf8ee\uf8f0(h
2
)2
\u2212 y2
\uf8f9\uf8fb
e
dA = bdy
para calcular a integral, ou calcule o volume do so´lido de tenso\u2dces usando a fo´rmula da
a´rea do segmento de para´bola.
103
Observac¸o\u2dces
1. Demonstra-se da Teoria da Elasticidade (Meca\u2c6nica dos so´lidos I) que a tensa\u2dco de
cisalhamento na\u2dco e´ exatamente constante na largura da sec¸a\u2dco, conforme a hipo´tese
ba´sica. Enta\u2dco a tensa\u2dco calculada e´ a tensa\u2dco me´dia na largura, enquanto que a tensa\u2dco
ma´xima e´ calculada na teoria da elasticidade. \u3c4med =
QMs
Izb
LN
y
\u3c4
 med
A
\u3c4
max
B
Figura 3.112: Tenso\u2dces cisalhante me´dia
A tabela abaixo (Beer-Johnstom, pa´g 276) ,mostra que o erro cometido varia com
a raza\u2dco b
h
b/h 1/4 1/2 1 2 4
\u3c4max/\u3c4med 1,008 1,033 1,126 1,396 1,988
diferenc¸a percentual 0,8% 3,3% 12,6% 39,6% 98,8%
2. Na realidade as sec¸o\u2dces permanecem planas, mas \u201cempenadas\u201d, pois a deformac¸a\u2dco
espec´\u131fica no cisalhamento e´ a distorc¸a\u2dco angular \u3b3 = \u3c4
G
.
Nos bordos livres (superior e inferior): \u3c4 = 0\u2192 \u3b3 = 0
\ufffd\ufffd
\ufffd\ufffd
\ufffd\ufffd
\ufffd\ufffd
\ufffd\ufffd
\ufffd\ufffd
\ufffd\ufffd
\ufffd\ufffd
\ufffd\ufffd
\ufffd\ufffd
\ufffd\ufffd
\ufffd\ufffd
\ufffd\ufffd
\ufffd\ufffd
\ufffd\ufffd
\ufffd\ufffd
\ufffd\ufffd
\ufffd\ufffd
Figura 3.113: Deformac¸a\u2dco cisalhante especifica nas bordas
Na Linha Neutra: \u3c4max \u2192 \u3b3max
Esta deformac¸a\u2dco, em um ca´lculo mais rigoroso, altera a ana´lise de tenso\u2dces e de-
formac¸o\u2dces na flexa\u2dco simples. No entanto, este efeito e´ desprezado, pois o erro
cometido e´ muito pequeno, exceto na regia\u2dco de aplicac¸a\u2dco de cargas concentradas.
3.4.3 Tenso\u2dces de Cisalhamento em Vigas de Sec¸a\u2dco de Diferentes
Formas
Admite-se a mesma hipo´tese ba´sica da sec¸a\u2dco retangular, isto e´, \u3c4 constante na largura da
sec¸a\u2dco. Obte´m-se as propriedades:
Tensa\u2dco de cisalhamento:
\u3c4 =
QMs
Izt
sendo t = t(y) e´ a largura (espessura) da camada considerada.
104
Sec¸o\u2dces T, I, caixa\u2dco, etc... (lados paralelos ou perpendiculares a` LN
Figura 3.114: Tipos de sec¸o\u2dces
1. Exemplos de sec¸a\u2dco T e I.
\u3c4
\u3c4
max
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
LN
e
b
b1
2
Figura 3.115: Sec¸a\u2dco T
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\u3c4
max
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
LN
e
b
\u3c4
Figura 3.116: Sec¸a\u2dco I
\u2022 Na mesa: O ca´lculo de \u3c4 esta´ sujeito a erro considera´vel ( b
h
grande), mas de
qualquer forma sa\u2dco tenso\u2dces pequenas.
\u2022 Na alma: O ca´lculo de \u3c4 produz resultados confia´veis, \u3c4max na LN.
\u2022 Na transic¸a\u2dco mesa-alma: descontinuidade no diagrama de tenso\u2dces.
2. Exemplo da figura 3.117. Sec¸a\u2dco retangular vazada (sec¸a\u2dco caixa\u2dco), ana´lise semelhante
a sec¸o\u2dces I, mas com \u3c4 = QMs
Iz(2e)
nas \u201calmas\u201d.
3.4.4 Exerc´\u131cios
1. Uma viga simplesmente apoiada em seus extremos tem 200 mm de largura por 400
mm de altura e 4 m de comprimento e suporta uma carga uniformemente distribu´\u131da
sobre todo seu comprimento. A tensa\u2dco longitudinal admiss´\u131vel e´ 12 MPa (trac¸a\u2dco e
compressa\u2dco) e a tensa\u2dco tangencial horizontal admiss´\u131vel e´ de 0,8 MPa. Determine o
valor ma´ximo admiss´\u131vel da carga por unidade de comprimento.
Resposta: q = 21,4 kN/m
105
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\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd
\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd