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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 \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd