resmat2007a
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Os produtores de perfis fornecem tabelas com as caracteristicas geome´tricas (dimensso\u2dces, a´rea, momento de ine´rcia...) necessa´rias ao projeto. Na Resiste\u2c6ncia dos Materiais I va- mos usar as tabelas do livro \u201cResiste\u2c6ncia dos Materiais\u201d de Beer e Johnston, que esta\u2dco reproduzidas em anexo. Os perfis sa\u2dco designados pela letra S(perfil I) ou W(perfil H) seguida da altura nominal (mm) e da sua massa em kg por metro (kg/m). Encontram-se em ordem decrescente de altura e, em cada grupo de mesma altura, em ordem decrescente de peso. 3.3.5 Exerc´\u131cios 1. Calcular o valor ma´ximo admissivel da carga P, na viga na figura 3.62 para uma \u3c3 = 140Mpa, se a viga e´ um perfil W150× 37, 1. Na\u2dco desprezar o peso pro´prio do perfil. Resposta: 14, 88 kN 2. Escolher o perfil I mais econo\u2c6mico para a viga da figura 3.63, para \u3c3 = 140Mpa Resposta: S 510× 97, 3 77 \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd P 2,5m Figura 3.62: Exerc´\u131cio 1 \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd 8m BA 27kN/m Figura 3.63: Exerc´\u131cio 2 3. Duplicando a carga da viga do exerc´\u131cio 2 (q\u2032 = 54 kN/m) e conservando o perfil adotado, para se obter resiste\u2c6ncia sa\u2dco soldados duas chapas (mesma \u3c3 = 140 MPa) sobre as mesas, de espessura do reforc¸o igual a espessura da mesa. Determine a largura das chapas e o trecho da viga em que e´ necessa´rio usa´-las. Desprezar os pesos pro´prios. Resposta: largura 121 mm, reforc¸o nos 5,0 m centrais da viga 4. A viga da figura 3.64 e´ contituida de um perfil W 200 × 86, de ac¸o com \u3c3 = 130 MPa). Calcular o valor ma´ximo admissivel de P desprezando o peso pro´prio. Resposta: 59, 57 kN/m \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd 5,4m A B Figura 3.64: Exerc´\u131cio 4 5. Calcular as tenso\u2dces extremas na viga da figura 3.65, indicando a sec¸a\u2dco onde ocorrem. A viga e´ constitu´\u131da por um perfil W130×28, 1. Considerar o efeito do peso pro´prio, ale´m da sobrecarga. Resposta: ±66, 1 MPa \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd 5,0m 1,5kN Figura 3.65: Exerc´\u131cio 5 78 6. Idem para a viga da figura 3.66 constitu´\u131da por um perfil W150x37, 1 Resposta: ±10, 77 MPa \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd 5,0m 1,5kN Figura 3.66: Exerc´\u131cio 6 7. Escolher o perfil mais econo\u2c6mico (I ou W, conforme indicado) para cada uma da figura 3.67, desconsiderando o efeito do peso pro´prio,ale´m da sobrecarga represen- tada. A tensa\u2dco admissivel e´ dada. \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \u3c3 \u3c3 \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \u3c3 = 140Mpa d) Perfil W,\u3c3 c) Perfil W, = 120Mpa \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd \ufffd 12kN 0,8m (S 130 x 15 ) 30kN 10kN/m 2,0m 2,0m (S 310 x 47,3) a) Perfil I, b) Perfil I, = 120Mpa = 140Mpa 65kN 65kN 1,0m 0,6m0,6m (W 250 x 32,7 ou W 310 x 32,7) 3,0m 25kN/m (W 460 x 52) Figura 3.67: Exerc´\u131cio 7 8. Para uma tensa\u2dco admiss´\u131vel de 150 MPa, calcular o valor ma´ximo admissivel de q na viga da figura 3.68, constitudida por duas chapas de ac¸o, 200 mm de largura e 12 mm de espessura, soldadas a dois perfis I (S 180× 30), conforme indicado na figura 3.68.Resposta: q = 27,05 kN/m 3.3.6 Vigas de dois materiais Sa\u2dco vigas de madeira reforc¸adas por cintas meta´licas, vigas de concreto reforc¸adas com barras de ac¸o (concreto armado), vigas-sanduiche, etc, genericamente designadas por vigas armadas. Estas vigas sa\u2dco constituidas por elementos longitudinais (camadas) de materiais difer- entes, seguramente aderentes de modo a ter necessa´ria resiste\u2c6ncia a`s tenso\u2dces tangenciais longitudinais Sa\u2dco admitidas as mesmas hipo´teses da flexa\u2dco em vigas de um so´ material. Portanto, para um momento fletor Mz = M , as sec¸o\u2dces permanecem planas e normais ao eixo e a 79 \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd q(kN/m) 6,0m0,6m 0,6m Figura 3.68: Exerc´\u131cio 8 deformac¸ ao especifica em uma camada de ordenada y em relac¸a\u2dco a LN (linha neutra) e´ ²x = ky (k constante) A figura 3.69 representam a sec¸a\u2dco transversal, o diagrama de deformac¸o\u2dces espec´\u131ficas e o diagrama de tenso\u2dces de uma viga constituida de dois materiais com a´reas de sec¸a\u2dco A1 e A2 e mo´dulos de elasticidade E1 e E2, respectivamente. Nestas figuras adimitimos E1 < E2 e a LN situada acima da superf´\u131cie de contato entre os materiais, mas as concluso\u2dces sa\u2dco gene´ricas. 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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 \ufffd\ufffd\ufffd \ufffd\ufffd\ufffd \ufffd\ufffd\ufffd \ufffd\ufffd\ufffd \ufffd\ufffd\ufffd \ufffd\ufffd\ufffd \ufffd\ufffd\ufffd \ufffd\ufffd\ufffd A , E M M E 2 2 \u3b5x =ky \u3c3x2 =E \u3b5x2 \u3c3x =E \u3b5x1 1 y L.N. M Figura 3.69: Viga de dois materiais Na camada de contato entre os dois materiais ha´ uma descontinuidade no diagrama de tenso\u2dces, com valores \u3c3x1 = E1²x para o material 1 e \u3c3x2 = E2²x para o material 2. A posic¸a\u2dco da LN e o valor da constante k sera\u2dco determinados