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Comparar as tenso\u2dces de cisalhamento e os a\u2c6ngulos de torc¸a\u2dco em dois tubos de paredes delgadas, um de parede delgadas,um de sec¸a\u2dco cirular e outro de sec¸a\u2dco quadrada, mas de mesmo comprimento, mesma a´rea de sec¸a\u2dco e mesma espessura, sujeitos ao mesmo torque. Resposta \u3c4circular \u3c4quadrada = 0, 7854 e \u3b8circular \u3b8quadrada = 0, 617 10. Uma chapa de ac¸o de 500mm de largura e 3mm de espessurae´ usada para fazer um tubo, curvando-se a chapa em 3600 e soltando-se as bordas juntas longitudinalmente (topo a topo). As formas a considerar sa\u2dco (a) circular, (b) quadrada (c) reta\u2c6ngular 150× 100mm .Admita um comprimento me´dio de 500mm (nenhum esforc¸o na placa devido ao encurvamento e cantos retos para sec¸o\u2dces, na\u2dco circulares). calcular o momento torque ma´ximo admissivel e o correspondente a\u2c6ngulo de torc¸a\u2dco para 2m de comprimento, em cada caso, dados G = 80Gpa e \u3c4 = 70Mpa. Resposta 8, 04kNm e 0, 0224rd; 6, 25kNm e 0, 0287rd; 5, 99kNm e 0, 0299rd 11. A figura 3.43 representa a sec¸a\u2dco tansversal da fuselagem de um avia\u2dco feito de liga de alum\u131´nio (G = 27 GPa). As espessuras das placas sa\u2dco 1,5 mm em AB e CD; 1,2 mm em BC e 1,0 mm em DA. Dados \u3c4 = 85 MPa, calcular o momento torsor admiss´\u131vel e o correspondente a\u2c6ngulo de torc¸a\u2dco. Resposta 124,59 kN e 0,00575 rad. 66 B C DA 700 mm 35 0 m m 50 0 m m 35 0 m m Figura 3.43: Figura do exerc´\u131cio 11 67 3.3 Solicitac¸a\u2dco por momento fletor 3.3.1 Introduc¸a\u2dco Uma barra de eixo reto e cargas transversais esta´ sujeita a momentos fletores e esforc¸os cortantes, geralmente. A barra e´ designada por viga e o efeito do momento fletor e´ a flexa\u2dco. 1. Tipos de Flexa\u2dco ( de acordo com os esforc¸os simples atuantes) PURA : somente momento fletor, constante (esforc¸o cortante nulo); SIMPLES : momento fletor e esforc¸o cortante; COMPOSTA : momento fletor e esforc¸o normal. \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd ~Ex. 1) Flexao Pura DMF constante DEC nulo + A B BA DMF M0 P P A B a L a P.a DMF A B DC DECP P DC A B Ex. 2) Flexao pura no vao AB Flexao simples nos balancos ~ ~ ~ ~ Figura 3.44: Tipos de Flexa\u2dco 2. Tipos de Flexa\u2dco (de acordo com os mom. fletores atuantes) Os eixos locais y e z sa\u2dco os eixos principais de ine´rcia da sec¸a\u2dco. O eixo de solicitac¸a\u2dco (ES) e´ a intersec¸a\u2dco do plano de solicitac¸a\u2dco (que conte´m as cargas e reac¸o\u2dces) com a sec¸a\u2dco transversal. 1. Flexa\u2dco Normal ou Reta: \u2022 ES \u2261 eixo -y \u2192My = 0 e Mf =Mz (ex.3 e 4) \u2022 ES \u2261 eixo -z \u2192Mz = 0 e Mf =My 2. Flexa\u2dco obl´\u131qua: ES na\u2dco coincide com nenhum dos eixos principais de ine´rcia: \u2212\u2192 M f = \u2212\u2192 My + \u2212\u2192 M z (ex. 5 e 6) 68 Observac¸o\u2dces: 1. Mf =M e´ normal ao plano de solicitac¸a\u2dco, portanto M \u22a5 ES 2. Res Mat.I \u2192 Flexa\u2dco Reta, Pura e Simples Res Mat.II \u2192 Flexa\u2dco obl´\u131qua e flexa\u2dco composta \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd 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E.S. z z y = E.S. y = E.S. y z z y Ex.3) Ex. 4) Ex. 5) Ex. 6) Mz M=Mz G My G M=Mz G Figura 3.45: Direc¸o\u2dces de Momentos Fletores e de eixos de solicitac¸o\u2dces 3.3.2 Ca´lculo das Tenso\u2dces Normais 1. Flexa\u2dco Pura e Reta Para vigas horizontais ou com pequena inclinac¸a\u2dco, admitindo-se pequenas deformac¸o\u2dces ela´sticas e proporcionais, sendo va´lida portanto a Lei de Hooke: \u3c3x = E²x Seja a viga AB do ex.1 anterior, fazendo M0 =M(M > 0) L A B A B comp < L comp > L M M Figura 3.46: Configurac¸o\u2dces inicial e deformada de uma viga bi-apoias sob flexa\u2dco pura. - Linhas longitudinais (fibras longitudinais ao eixo) assumem o aspecto curvo. O eixo deformado a` flexa\u2dco e´ a linha ela´stica . - Linhas transversais (sec¸o\u2dces transversais) permanecem retas (planas) e \u22a5s ao eixo deformado. Sofrem um rotac¸a\u2dco em torno do eixo-z local. - Uma camada de fibras situadas em um plano horizontal na configurac¸a\u2dco inicial mante´m o comprimento L ( ²x = 0\u2192 \u3c3x = 0). E´ designada por superf´\u131cie neutra e sua intersec¸a\u2dco