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qstruct:teoria:qsection:risultante_sezione_elastica

Risultante sezione in materiale elastico

Consideriamo una sezione composta da un materiale elastico-lineare, soggetta ad uno stato deformativo piano del tipo

$$\varepsilon = \lambda + \mu_z \; y + \mu_y \; z $$

Assumendo una relazione elastico lineare tra tensione e deformazioni abbiamo

$$N = \iint \limits_S \sigma \; \mathrm{d}y \mathrm{d}z = \iint \limits_S E \left( \lambda + \mu_z \; y + \mu_y \; z \right) \; \mathrm{d}y \mathrm{d}z = E \left( \lambda \iint \limits_S \; \mathrm{d}y \mathrm{d}z + \mu_z \iint \limits_S y \; \mathrm{d}y \mathrm{d}z + \mu_y \iint \limits_S z \; \mathrm{d}y \mathrm{d}z \right) \Longrightarrow \; \\ \; \Longrightarrow N = E \left( \lambda \; A + \mu_z \; S_z + \mu_y \; S_y \right) $$

$$M_y = \iint \limits_S \sigma \; z \; \mathrm{d}y \mathrm{d}z = \iint \limits_S E \left( \lambda + \mu_z \; y + \mu_y \; z \right) z \; \mathrm{d}y \mathrm{d}z = E \left( \lambda \iint \limits_S z \; \mathrm{d}y \mathrm{d}z + \mu_z \iint \limits_S y \; z \; \mathrm{d}y \mathrm{d}z + \mu_y \iint \limits_S z^2 \; \mathrm{d}y \mathrm{d}z \right) \Longrightarrow \; \\ \; \Longrightarrow M_y = E \left( \lambda \; S_y + \mu_z \; I_{yz} + \mu_y \; I_{yy} \right) $$

$$M_z = - \iint \limits_S \sigma \; y \; \mathrm{d}y \mathrm{d}z = - \iint \limits_S E \left( \lambda + \mu_z \; y + \mu_y \; z \right) y \; \mathrm{d}y \mathrm{d}z = - E \left( \lambda \iint \limits_S y \; \mathrm{d}y \mathrm{d}z + \mu_z \iint \limits_S y^2 \; \mathrm{d}y \mathrm{d}z + \mu_y \iint \limits_S y \; z \; \mathrm{d}y \mathrm{d}z \right) \Longrightarrow \; \\ \; \Longrightarrow M_z = - E \left( \lambda \; S_z + \mu_z \; I_{zz} + \mu_y \; I_{yz} \right) $$


qstruct/teoria/qsection/risultante_sezione_elastica.txt · Ultima modifica: 2016/09/27 13:21 da mickele

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