Uma viga em equilíbrio, no regime elástico, está sob exão oblíqua tal que a inclinação do momento etor forme 50°com o eixo z. Seja a seção reta u...

Para encontrar a inclinação da linha neutra, podemos utilizar a equação de Navier-Bernoulli para vigas em flexão: M = -EI(d²y/dx²) Onde: M = momento fletor E = módulo de elasticidade I = momento de inércia da seção transversal y = deslocamento da linha neutra em relação ao eixo x Podemos reescrever a equação acima como: d²y/dx² = -M/(EI) Sabemos que a intensidade do momento fletor é de 2.000 N.m e que a seção transversal é um círculo de raio 300 mm, o que significa que o momento de inércia é igual a (π/4)*(300 mm)⁴ = 1,413 x 10¹⁰ mm⁴ (considerando que o material da viga é homogêneo e isotrópico). Assumindo que a viga está em equilíbrio, podemos concluir que a soma das forças verticais e horizontais é igual a zero. Como a viga está sob flexão oblíqua, a componente horizontal do momento fletor é igual a M*cos(50°) = 1.282,5 N.m e a componente vertical é igual a M*sin(50°) = 1.538,8 N.m. Substituindo os valores na equação de Navier-Bernoulli, temos: d²y/dx² = -1.282,5 N.m/(200 GPa * 1,413 x 10¹⁰ mm⁴) = -0,00000455 mm⁻² Integrando a equação acima duas vezes, obtemos: y = (-1,2825/6EI)x³ + C1x + C2 Como a viga está em equilíbrio, sabemos que a linha neutra passa pelo centro de gravidade da seção transversal, que no caso é o centro do círculo. Portanto, podemos assumir que y = 0 quando x = 0. Além disso, como a inclinação da linha neutra forma um ângulo de 50° com o eixo z, podemos concluir que dy/dx = tan(50°) = 1,1918. Substituindo esses valores na equação acima, temos: 0 = 0 + C1*0 + C2 1,1918 = (-1,2825/2EI)*0² + C1*0 + C2 + C1 Portanto, C2 = 0 e C1 = 1,1918. Assim, a equação da linha neutra é: y = (-1,2825/6EI)x³ + 1,1918x A inclinação da linha neutra em relação ao eixo z é dada por: dy/dx = (-1,2825/2EI)x² + 1,1918 Podemos calcular a tangente da inclinação substituindo x pelo raio da seção transversal: dy/dx = (-1,2825/2EI)*(300 mm)² + 1,1918 = 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