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qstruct:teoria:qsection:risultante_sezione_in_cls

Risultante sezione in calcestruzzo

Legge parabola rettangolo

$$\sigma_c = \begin{cases} \frac{f_{cd}}{\varepsilon_{c2}^2} \varepsilon_c^2 + 2 \frac{f_{cd}}{\varepsilon_{c2}} \varepsilon_{c} & - \varepsilon_{c2} \le \varepsilon_c \le 0 \\\\ - f_{cd} & - \varepsilon_{cu2} \le \varepsilon_c \le \varepsilon_{c2} \end{cases} $$

Calcolo risultante mediante integrazione

$$N_{Rd} = \iint \limits_{S1} \left[ \frac{f_{cd}}{\varepsilon_{c2}^2} \left( \lambda + \mu_y z + \mu_z y \right)^2 + 2 \frac{f_{cd}}{\varepsilon_{c2}} \left( \lambda + \mu_y z + \mu_z y \right) \right] \mathrm{d}y \mathrm{d}z - f_{cd} \iint \limits_{S2} \mathrm{d}y \mathrm{d}z = \\ \iint \limits_{S1} \left[ \frac{f_{cd}}{\varepsilon_{c2}^2} \left( \lambda^2 + \mu_y^2 z^2 + \mu_z^2 y^2 + 2 \lambda \mu_y z + 2 \lambda \mu_z y + 2 \mu_y \mu_z y z \right) + 2 \frac{f_{cd}}{\varepsilon_{c2}} \left( \lambda + \mu_y z + \mu_z y \right) \right] \mathrm{d}y \mathrm{d}z - f_{cd} \iint \limits_{S2} \mathrm{d}y \mathrm{d}z = \\ \left[ \frac{f_{cd}}{\varepsilon_{c2}^2} \left( \lambda^2 A_1 + \mu_y^2 I_{yy,1} + \mu_z^2 I_{zz,1} + 2 \lambda \mu_y S_{y,1} + 2 \lambda \mu_z S_{z,1} + 2 \mu_y \mu_z I_{yz,1} \right) + 2 \frac{f_{cd}}{\varepsilon_{c2}} \left( \lambda A_1 + \mu_y S_{y,1} + \mu_z S_{z,1} \right) \right] - f_{cd} A_2 $$

$$M_{Rd,y} = \iint \limits_{S1} \left[ \frac{f_{cd}}{\varepsilon_{c2}^2} \left( \lambda + \mu_y z + \mu_z y \right)^2 + 2 \frac{f_{cd}}{\varepsilon_{c2}} \left( \lambda + \mu_y z + \mu_z y \right) \right] z \mathrm{d}y \mathrm{d}z - f_{cd} \iint \limits_{S2} z \mathrm{d}y \mathrm{d}z = \\ \iint \limits_{S1} \left[ \frac{f_{cd}}{\varepsilon_{c2}^2} \left( \lambda^2 z + \mu_y^2 z^3 + \mu_z^2 y^2 z + 2 \lambda \mu_y z^2 + 2 \lambda \mu_z y z + 2 \mu_y \mu_z y z^2 \right) + 2 \frac{f_{cd}}{\varepsilon_{c2}} \left( \lambda z + \mu_y z^2 + \mu_z y z \right) \right] \mathrm{d}y \mathrm{d}z - f_{cd} \iint \limits_{S2} z \mathrm{d}y \mathrm{d}z = \\ \left[ \frac{f_{cd}}{\varepsilon_{c2}^2} \left( \lambda^2 S_{y,1} + \mu_y^2 \iint \limits_{S1} z^3 \mathrm{d}y \mathrm{d}z + \mu_z^2 \iint \limits_{S1} y^2 z \mathrm{d}y \mathrm{d}z + 2 \lambda \mu_y I_{yy,1} + 2 \lambda \mu_z I_{yz,1} + 2 \mu_y \mu_z \iint \limits_{S1} y z^2 \mathrm{d}y \mathrm{d}z \right) + 2 \frac{f_{cd}}{\varepsilon_{c2}} \left( \lambda S_{y,1} + \mu_y I_{yy,1} + \mu_z I_{yz,1} \right) \right] - f_{cd} S_{y,2} $$

$$- M_{Rd,z} = \iint \limits_{S1} \left[ \frac{f_{cd}}{\varepsilon_{c2}^2} \left( \lambda + \mu_y z + \mu_z y \right)^2 + 2 \frac{f_{cd}}{\varepsilon_{c2}} \left( \lambda + \mu_y z + \mu_z y \right) \right] y \mathrm{d}y \mathrm{d}z - f_{cd} \iint \limits_{S2} y \mathrm{d}y \mathrm{d}z = \\ \iint \limits_{S1} \left[ \frac{f_{cd}}{\varepsilon_{c2}^2} \left( \lambda^2 y + \mu_y^2 y z^2 + \mu_z^2 y^3 + 2 \lambda \mu_y y z + 2 \lambda \mu_z y^2 + 2 \mu_y \mu_z y^2 z \right) + 2 \frac{f_{cd}}{\varepsilon_{c2}} \left( \lambda y + \mu_y y z + \mu_z y^2 \right) \right] \mathrm{d}y \mathrm{d}z - f_{cd} \iint \limits_{S2} y \mathrm{d}y \mathrm{d}z = \\ \left[ \frac{f_{cd}}{\varepsilon_{c2}^2} \left( \lambda^2 S_{z,1} + \mu_y^2 \iint \limits_{S1} y z^2 \mathrm{d}y \mathrm{d}z + \mu_z^2 \iint \limits_{S1} y^3 \mathrm{d}y \mathrm{d}z + 2 \lambda \mu_y I_{yz,1} + 2 \lambda \mu_z I_{zz,1} + 2 \mu_y \mu_z \iint \limits_{S1} y^2 z \mathrm{d}y \mathrm{d}z \right) + 2 \frac{f_{cd}}{\varepsilon_{c2}} \left( \lambda S_{z,1} + \mu_y I_{yz,1} + \mu_z I_{zz,1} \right) \right] - f_{cd} S_{z,2} $$

Legge bilineare

$$\sigma_c = \begin{cases} f_{cd} \frac{\varepsilon_{c}}{\varepsilon_{c3}} & - \varepsilon_{c3} \le \varepsilon_c \le 0 \\\\ - f_{cd} & - \varepsilon_{cu3} \le \varepsilon_c \le \varepsilon_{c3} \end{cases} $$

Calcolo risultante mediante integrazione

$$N_{Rd} = \iint \limits_{S1} \left[ \frac{f_{cd}}{\varepsilon_{c3}} \left( \lambda + \mu_y z + \mu_z y \right) \right] \mathrm{d}y \mathrm{d}z - f_{cd} \iint \limits_{S2} \mathrm{d}y \mathrm{d}z = \\ \left[ \frac{f_{cd}}{\varepsilon_{c3}} \left( \lambda A_1 + \mu_y S_{y,1} + \mu_z S_{z,1} \right) \right] - f_{cd} A_2 $$

$$M_{Rd,y} = \iint \limits_{S1} \left[ \frac{f_{cd}}{\varepsilon_{c3}} \left( \lambda + \mu_y z + \mu_z y \right) \right] z \mathrm{d}y \mathrm{d}z - f_{cd} \iint \limits_{S2} z \mathrm{d}y \mathrm{d}z = \\ \left[ \frac{f_{cd}}{\varepsilon_{c3}} \left( \lambda S_{y,1} + \mu_y I_{yy,1} + \mu_z I_{yz,1} \right) \right] - f_{cd} S_{y,2} $$

$$- M_{Rd,z} = \iint \limits_{S1} \left[ \frac{f_{cd}}{\varepsilon_{c3}} \left( \lambda + \mu_y z + \mu_z y \right) \right] y \mathrm{d}y \mathrm{d}z - f_{cd} \iint \limits_{S2} y \mathrm{d}y \mathrm{d}z = \\ \left[ \frac{f_{cd}}{\varepsilon_{c3}} \left( \lambda S_{z,1} + \mu_y I_{yz,1} + \mu_z I_{zz,1} \right) \right] - f_{cd} S_{z,2} $$

Stress block

$$\sigma_c = \begin{cases} 0 & - \varepsilon_{c4} \le \varepsilon_c \le 0 \\\\ - f_{cd} & - \varepsilon_{cu3} \le \varepsilon_c \le \varepsilon_{c4} \end{cases} $$

Calcolo risultante mediante integrazione

$$N_{Rd} = - f_{cd} \iint \limits_{S2} \mathrm{d}y \mathrm{d}z = - f_{cd} A_2 $$

$$M_{Rd,y} = - f_{cd} \iint \limits_{S2} z \mathrm{d}y \mathrm{d}z = - f_{cd} S_{y,2} $$

$$- M_{Rd,z} = - f_{cd} \iint \limits_{S2} y \mathrm{d}y \mathrm{d}z = - f_{cd} S_{z,2} $$


qstruct/teoria/qsection/risultante_sezione_in_cls.txt · Ultima modifica: 2016/10/14 15:58 da mickele

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